SlideShare a Scribd company logo
S E T S
A set is an unordered collection of
different elements. A set can be
written explicitly by listing its
elements using set bracket. If the
order of the elements is changed or
any element of a set is repeated, it
does not make any changes in the
set.
Some Example of Sets
● A set of all positive integers
● A set of all the planets in the solar system
● A set of all the states in India
● A set of all the lowercase letters of the alphabet
Cardinality of a Set
● Cardinality of a set S, denoted by |S|, is the number of elements of the set. The number is
also referred as the cardinal number. If a set has an infinite number of elements, its
cardinality is ∞.
Example − |{1,4,3,5}|=4,|{1,2,3,4,5,…}|=∞
TYPES OF SETS
Sets can be classified into many types. Some of which are finite,
infinite, subset, universal, proper, singleton set, etc.
Finite Set
● A set which consists of a definite number of elements is called a finite set.
Example: A set of natural numbers up to 10.
A = {1,2,3,4,5,6,7,8,9,10}
Empty Set
● A set which does not contain any element is called an empty set or void set or null set. It
is denoted by { } or Ø.
Example
● A set of apples in the basket of grapes is an example of an empty set because in a grapes
basket there are no apples present.
Singleton Set
● A set which contains a single element is called a singleton set.
Example: There is only one apple in the basket.
Example 2:
A = {10}
Infinite Set
● A set which is not finite is called an infinite set. We use ellipsis to indicate an infinite set.
● Example: A set of all natural numbers.
● A = {1,2,3,4,5,6,7,8,9……}
Equivalent set
● If the number of elements is the same for two different sets, then they are called equivalent
sets. The order of sets does not matter here. It is represented as:
n(A) = n(B)
where A and B are two different sets with the same number of elements.
Example: If A = {1,2,3,4} and B = {Red, Blue, Green, Black}
In set A, there are four elements and in set B also there are four elements. Therefore, set A and
set B are equivalent.
Equal sets
● The two sets A and B are said to be equal if they have exactly the same elements, the
order of elements do not matter.
Example: A = {1,2,3,4} and B = {4,3,2,1}
A = B
Disjoint Sets
● The two sets A and B are said to be disjoint if the set does not contain any common
element.
Example:
Set A = {1,2,3,4} and set B = {5,6,7,8} are disjoint sets, because there is no common element
between them.
Subsets
● A set ‘A’ is said to be a subset of B if every element of A is also an element of B, denoted
as A ⊆ B. Even the null set is considered to be the subset of another set. In general, a
subset is a part of another set.
Example: A = {1,2,3}
Then {1,2} ⊆ A.
Similarly, other subsets of set A are: {1},{2},{3},{1,2},{2,3},{1,3},{1,2,3},{}.
Note: The set is also a subset of itself.
If A is not a subset of B, then it is denoted as A⊄B.
Proper Subset
● If A ⊆ B and A ≠ B, then A is called the proper subset of B and it can be written as A⊂B.
Example: If A = {2,5,7} is a subset of B = {2,5,7} then it is not a proper subset of B = {2,5,7}
But, A = {2,5} is a subset of B = {2,5,7} and is a proper subset also.
Superset
A superset can be defined as a set of elements containing all of the elements of another set.
Set A is said to be the superset of B if all the elements of set B are the elements of set A. It is
represented as A ⊃ B.
For example, if set A = {1, 2, 3, 4} and set B = {1, 3, 4}, then set A is the superset of B.
Universal Set
● A set which contains all the sets relevant to a certain condition is called the universal set.
It is the set of all possible values.
Example: If A = {1,2,3} and B {2,3,4,5}, then universal set here will be:
U = {1,2,3,4,5}
SET OPERATIONS
1. Intersection (denoted by ∩): The intersection of two sets A and B is the set of all elements
that are in both A and B.
For example:
if A = {1, 3, 8} and B = {-9, 22, 3}, then A ∩ B = {3}1.
2. Disjoint Sets
● Two sets are disjoint if they have no elements in common1. In other words, A and B are
disjoint if their intersection is the empty set (∅)1.
3. Union (denoted by ∪)
● The union of two sets A and B is the set of all elements that are in A or in B or in both1.
For example, if A = {2, 5, 8} and B = {7, 5, 22}, then A ∪ B = {2, 5, 8, 7, 22}1.
4. Complement of a Set
● The complement of a set A (denoted by A′) is the set of elements which are not in set A.
5. Set Difference/ Relative Complement
● The set difference of sets A and B (denoted by A–B) is the set of elements which are only
in A but not in B.
Example − If A={10,11,12,13} and B={13,14,15}, then (A−B)={10,11,12} and (B−A)={14,15}.
Here, we can see (A−B)≠(B−A)
Power Set
● Power set of a set S is the set of all subsets of S including the empty set. The cardinality of a power
set of a set S of cardinality n is 2n. Power set is denoted as P(S).
Example −
For a set S={a,b,c,d} let us calculate the subsets −
Subsets with 0 elements − {∅}
(the empty set)
Subsets with 1 element − {a},{b},{c},{d}
Subsets with 2 elements − {a,b},{a,c},{a,d},{b,c},{b,d},{c,d}
Subsets with 3 elements − {a,b,c},{a,b,d},{a,c,d},{b,c,d}
Subsets with 4 elements − {a,b,c,d}
● Hence, P(S)=
● {{∅},{a},{b},{c},{d},{a,b},{a,c},{a,d},{b,c},{b,d},{c,d},{a,b,c},{a,b,d},{a,c,d},{b,c,d},{a,b,c,d}}
● |P(S)|=24=16
● Note − The power set of an empty set is also an empty set.
● |P({∅})|=20=1
SET RELATIONS
WHAT IS SET RELATIONS?
 A set relation is a fundamental concept in mathematics that allows us to
describe and understand the connections or associations between elements
within sets. In the context of set theory, a relation is essentially a set of
ordered pairs.
 Set relations play a crucial role in various fields, including mathematics,
computer science, data analysis, and beyond. They provide a foundation for
understanding relationships, making comparisons, and solving problems.
Significance:
● Understanding Relationships
● Comparisons and Classification
● Problem solving
● Foundation for further concepts
 A set of ordered pairs is defined as a
relation.’
 This mapping depicts a relation from set A
into set B. A relation from A to B is a
subset of A x B. The ordered pairs are
(1,c),(2,n),(5,a),(7,n). For defining a
relation, we use the notation where,
• set {1, 2, 5, 7} represents the domain.
• set {a, c, n} represents the range.
Finding the domain, range, and codomain of a relation using the roster method
involves identifying the elements that belong to each of these sets based on the
ordered pairs in the relation. Let's break down how to find each of these components
using an example.
Example Relation: Consider the relation R between set A = {1, 2, 3} and set B
= {4, 5, 6} defined by the following ordered pairs: R = {(1, 4), (2, 5), (3, 6)}
Domain:
The domain of a relation consists of all the first elements (the elements from
the left side of the ordered pairs) in the relation.
In this case, the domain of R is the set of all first elements of the ordered pairs
in R.
Domain(R) = {1, 2, 3}
Range:
The range of a relation consists of all the second elements (the elements from
the right side of the ordered pairs) in the relation.
In this case, the range of R is the set of all second elements of the ordered pairs
in R.
Range(R) = {4, 5, 6}
Codomain:
The codomain is the set that specifies the possible values for the second
elements of the ordered pairs in the relation. It is predetermined and doesn't
depend on the actual ordered pairs in the relation.
In this case, the codomain of R is set B, which is {4, 5, 6}.
So, for the given relation R using the roster method:
Domain(R) = {1, 2, 3,}
Range(R) = {4, 5, 6}
Codomain(R) = {4, 5, 6}
9
8
TYPES OF SET RELATIONS
Empty Relation
1
2
3
4
6
7
Full Relation
Identity Relation
Inverse Relation
Symmetric Relation
Transitive Relation
Equivalence Relation
Partial Order Relation
5 Reflexive Relation
10
Anti-Symmetric Relation
 An empty relation (or void relation) is one in
which there is no relation between any elements
of a set. It is one of the simplest types of set
relations.
For empty relation, R = φ ⊂ A × A
 In this case, there are no ordered pairs in the
relation R, which means there is no connection
or relationship between any element.
EMPTY RELATION
• Set X: {1, 3, 5}
• Set Y: {2, 4, 6}
R={ };
This is a clear example of an empty
relation, signifying the absence of any
association or interaction between the two
sets.
FULL RELATIONS
A Full relation is a type of relation in which
every element of a set is related to each other,
also known as the universal relation or complete
relation, which is the opposite of an empty
relation. It represents a set relation where every
possible pair of elements from two sets is
included.
The full relation is often denoted as U (for
universal) or sometimes as the set of all possible
pairs of elements, denoted as U = {(a, b) | a ∈
Set A, b ∈ Set B}.
Example:
Consider two sets:
● Set A: {Alice, Bob, Carol}
● Set B: {X, Y}
The full relation between these
sets would be represented as a
set of ordered pairs:
R = {(Alice, X), (Alice, Y), (Bob,
X), (Bob, Y), (Carol, X), (Carol,
Y)}
In this relation R, every possible
pair of elements from Set A and
Set B is included, resulting in a
complete and exhaustive set of
connections.
Imagine a set of ordered pairs that connects every
element in Set A to every element in Set B,
forming a complete web of connections.
The formula for a full relation R between two sets
A and B is typically represented as: R = A × B
• Here, "×" represents the Cartesian product of
sets A and B, which generates all possible
ordered pairs of elements from A and B.
Therefore, R includes all the ordered pairs,
making it the full relation between A and B.
Example 2:
Consider two sets:
● Set A: {1,4, 8l}
● Set B: {x, y}
The full relation between these sets would be
represented as a set of ordered pairs:
R = {(1, X), (1, Y), (4, X), (4, Y), (8, X), (8, Y)}
In this relation R, every possible pair of
elements from Set A and Set B.
IDENTITY RELATIONS
 Set A is a relation where every element of A is
related to itself only. This means that if you
have a set “A”, the identity relation “I” contains
pairs of elements from “A” where both elements
in each pair are the same. This relation reflects
the concept of self-identity or equality within the
set.
The condition for the identity relation on set A is
represented as: I = {(a, a) | a ∈ A}
In this condition: (a, a) represents an ordered
pair where an element "a" is related to itself.
"a ∈ A" indicates that "a" is an element of set A.
Example: set A={a, b, c}, the
identity relation will be
I = {a, a},{b, b}, {c, c}
For Identity relation
I = {(a, a), a ∈ A}
Identity relation consists of all such ordered pairs where each element in set A is
related only to itself.
Example:
Consider a set A that represents the set of natural numbers less than or equal to 5:
A = {1, 2, 3, 4, 5}
The identity relation (Id_A) on this set A would consist of ordered pairs where each
element is related to itself. Here's what the identity relation would look like for this
set:
Id_A = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)}
In this example, every element from the set A is related only to itself, which is the
defining characteristic of the identity relation.
INVERSE RELATION
Inverse relation is seen when a set has elements
which are inverse pair of another set.
If (x, y) ∈ R, then (y, x) ∈ R^(-1) and vice versa.
i.e., If R is from A to B, then R^(-1) is from B to A.
Thus, if R is a subset of A x B, then R-1 is a
subset of B x A.
An inverse relation is the inverse of a relation
and is obtained by interchanging the elements of
each ordered pair of the given relation.
Example:
Have a look at the following relations and their inverse relations on two sets A = {a, b, c, d, e}
and B = {1, 2, 3, 4, 5}.
• If R = {(a, 2), (b, 4), (c, 1)} ⇔ R-1 = {(2, a), (4, b), (1, c)}
• If R = {(c, 1), (b, 2), (a, 3)} ⇔ R-1 = {(1, c), (2, b), (3, a)}
• If R = {(b, 3), (c, 2), (e, 1)} ⇔ R-1 = {(3, b), (2, c), (1, e)}
The formula for the inverse relation R^(-1) of a relation R is defined as:
R^(-1) = {(b, a) | (a, b) ∈ R}
In this formula: (a, b) represents an ordered pair in the original relation R.
(b, a) represents the corresponding ordered pair in the inverse relation R^(-1).
So, R^(-1) contains all the ordered pairs from R, but with their elements reversed in order.
A relation R on a set A is said to be
reflexive if, for every element a in set A,
(a, a) belongs to R. In other words, every
element is related to itself.
Reflexive relations often represent
properties of elements that are inherent to
the elements themselves. They are
symbolically represented as (a, a) ∈ R for
all a in A.
REFLEXIVE RELATIONS
Where a is the element, A is the set and R
is the relation.
Example: (a, a) ∈ R
A = {1, 2, 3}
R1 = {(1, 1), (1, 2)} not Reflexive
R2 = {(1, 1), (1, 2)} Reflexive
R3 = {(1, 1), (2, 2), (3, 3)} Reflexive
RA = {(1, 1), (2, 2), (3, 2)} not
Reflexive
R5 = { } not Reflexive
R6 = A x A Reflexive
How to calculate how many Reflexive
relations are there if we have an n element.
N=
Example: A= {1,2,3}
N = 2(9-3)
N = 26
N = 64
A symmetric relation is a fundamental
concept in discrete mathematics that
describes a specific type of binary relation
between elements in a set. It possesses a key
property known as symmetry, which means
that if an element 'a' is related to an element
'b,' then 'b' is also related to ‘a.’
Condition for symmetric relation:
(a,b) ∈ R ⇒ (b,a) ∈ R for all a, b ∈ A.
aRb ⇒ bRa for all a,b ∈ A.
SYMMETRIC RELATION
Example:
Consider a set of cities and a relation R defined on pairs of cities. We define R as follows:
(City A, City B) is in relation R if and only if there is a direct flight between City A and City
B.
Demonstrating Symmetry:
• Suppose there is a direct flight from City A to City B. According to the definition, (City
A, City B) ∈ R.
• Now, since there is a direct flight, it also implies that there is a direct flight from City
B to City A.
• Therefore, (City B, City A) must also be in relation R. This example illustrates that if
there's a direct flight from A to B, there's also a direct flight from B to A, satisfying
the symmetry property.
Symmetric relations often appear in real-world scenarios, such as transportation networks.
In this example, the symmetric relation R reflects the bi-directional nature of direct flights
between cities.
Symmetric Relation Formula:
The number of symmetric
relations on a set with ‘n’
elements is given by the
formula:
N =
Example: A={1,2,3}
N= 2n(n+1)/2
N = 2(3)(3+1)/2
= 2(3)(4)/2
= 212/2
= 26
= 64
Example 1: Suppose R is a relation on a set P where A = {3, 4,
5} and R = {(3,3), (3,4), (3,5), (4,5), (5,3)}. Check if R is a
symmetric relation.
Solution: As we can view that (3,4) ∈ R. For R to be symmetric
(4, 3) should be in R although (4, 3) ∉ R.Also (4,5)∈ R but (5, 4)
∉ R
Therefore, R is not a symmetric relation.
Example 2: Let Z be the set of two female kids (z, x) in a
family and R be a relation.
Solution: Let z, x ∈ Z.
If “z” is the sister of “x”, then “x” has to be the sister of “z”.
We can say that, R = {(z, x), (x, z)}
So, R is symmetric.
 A relation R on a set A is said to be asymmetric if and only if (a,b)∈R , then
(b,a)∉R , for all a,b∈A. In other words, an asymmetric relation is the
opposite of a symmetric relation.
Example: The relation R “is a parent of a and b” is asymmetric since if a is the
parent of b , then b cannot be the parent of a
 A relation R on a set A is known as asymmetric relation if no (b,a) ∈ R
when (a,b) ∈ R or we can even say that relation R on set A is symmetric if
only if (a,b) ∈ R⟹(b, a) ∉R.
ASYMMETRIC RELATION
Anti-symmetric Relation:
A relation R on a set A is said to be antisymmetric, if aRb and bRa
holds if and only if when a=b. In other words, (a,b)∉R and (b,a)∉R
if a≠b.
Example:
Let us consider A to be the set on which the relation R is defined,
then R is said to be antisymmetric when aRb and bRa⇒a=b
where a, b∈A
i.e. If (a, b)∈R & (b, a)∈R, then a=b. where, a∈A and b∈B.
ANTI-SYMMETRIC RELATION
 Transitive relations are binary relations in set
theory that are defined on a set A such that if a is
related to b and b is related to c, then element a
must be related to element c, for a, b, c in set A.
 A binary relation R defined on a set A is said to be
a transitive relation for all a, b, c in A if a R b and
b R c, then a R c, that is, if a is related to b and b is
related to c, then a must be related to c.
Mathematically, we can write it as: a relation R
defined on a set A is a transitive relation for all a,
b, c ∈ A, if (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R.
TRANSITIVE RELATION
 Example 1:
 Define a relation R on a set A = {a, b, c} as R = {(a, b), (b, c), (b, b)}. Determine if R is a
transitive relation.
 Solution: As we can see that (a, b) ∈ R and (b, c) ∈ R, and for R to be transitive (a, c) ∈ R
must hold, but (a, c) ∉ R. So, R is not a transitive relation.
 Answer: R is not a transitive relation
 Example 2:
Consider A ={1, 2, 3, 4}
R1={(1,1), (1,2), (2,3), (1,3), (4,4) Transitive
R2={(1,1), (1,2), (2,1), (2,2), (3,3), (4,4) Transitive
R3={(1,3), (2,1)} Not Transitive
EQUIVALENCE RELATION
 An equivalence relation is a binary relation
defined on a set X such that the relation is
reflexive, symmetric and transitive. If any of
the three conditions (reflexive, symmetric and
transitive) does not hold, the relation cannot
be an equivalence relation. The equivalence
relation divides the set into disjoint
equivalence classes. Any two elements of the
set are said to be equivalent if and only if they
belong to the same equivalence class.
 A relation R defined in a
set is called an
Equivalence relation if
it satisfy the following:
Example2:
Consider a group of friends who are trying to determine if they have similar tastes in music. They decide to
categorize their music preferences based on whether they have the same favorite genre.
Defining the Relation:
Let's represent each friend by a letter: A, B, C, D, and so on.
We say that two friends, denoted as (a, b) are in the same group if they share the same favorite music genre.
Demonstrating Equivalence:
• Reflexivity: Each person, like A, has themselves as their friend (a, a). So, everyone is in their own group,
indicating that they share the same taste as themselves.
• Symmetry: If A and B are in the same group, it means they have the same favorite genre. Therefore, B and A
are also in the same group, showing that the relationship is mutual.
• Transitivity: If A and B are in the same group (same music taste), and B and C are also in the same group,
then it follows that A and C share the same favorite genre. So, the transitivity property holds.
• Equivalence Classes: In this example, each group consists of friends who share the same favorite music genre.
For instance, there may be a group of friends who all love rock music, another group who adore jazz, and so on.
 Also known as partially ordered sets or posets, are
a fundamental concept in mathematics and
computer science. They are a specific type of binary
relation that satisfies three key properties:
• Reflexivity: For every element 'a' in the set, 'a' is
related to itself. This is represented as: a ≤ a.
• Antisymmetry: If 'a' is related to 'b' and 'b' is
related to 'a', then 'a' and 'b' must be the same
element. In other words, if a ≤ b and b ≤ a, then a =
b.
• Transitivity: If 'a' is related to 'b' and 'b' is related
to 'c', then 'a' is related to 'c'. This is represented as:
If a ≤ b and b ≤ c, then a ≤ c.
PARTIAL ORDER RELATIONS
 Symbols
• Partial Order Relations are
typically denoted using the
symbol '≤' (less than or equal
to) or other similar symbols
like '⊆' (subset) or '⊇'
(superset) depending on the
context. For instance:
a ≤ b denotes that element 'a' is
related to element 'b'.
A ⊆ B represents that set 'A' is a
subset of set 'B'.
Example 1:
Consider the set of natural numbers (N) and the relation '≤' defined on N, where 'a ≤ b' if
and only if 'a' is divisible by 'b' without a remainder.
• Reflexivity: For any natural number 'a', 'a' is divisible by itself without a remainder, so
'a ≤ a' holds.
• Antisymmetry: If 'a ≤ b' and 'b ≤ a', then 'a' and 'b' must be the same number. For
instance, if '4 ≤ 2' (since 4 is divisible by 2) and '2 ≤ 4' (since 2 is divisible by 4), then 'a
= b = 2’.
• Transitivity: If 'a ≤ b' and 'b ≤ c', then 'a ≤ c'. For example, if '8 ≤ 4' (since 8 is divisible
by 4) and '4 ≤ 2' (since 4 is divisible by 2), then '8 ≤ 2' (since 8 is divisible by 2).
This example illustrates how the relation '≤' on natural numbers satisfies the properties of
a partial order relation, making it a partial order relation on the set of natural numbers.
Example 2:
 Show whether the relation (x, y) ∈ R, if, x ≥ y defined on the set of integers is a partial order
relation.
Consider the set A = {1, 2, 3, 4} containing four integers. Find the relation for this set such as
R = {(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3), (1, 1), (2, 2), (3, 3), (4, 4)}.
• Reflexive: The relation is reflexive as for every a ∈ A. (a, a) ∈ R, i.e. (1, 1), (2, 2), (3, 3), (4, 4) ∈ R.
• Antisymmetric: The relation is antisymmetric as whenever (a, b) and (b, a) ∈ R, we have a = b.
• Transitive: The relation is transitive as whenever (a, b) and (b, c) ∈ R, we have (a, c) ∈ R.
Example: (4, 2) ∈ R and (2, 1) ∈ R, implies (4, 1) ∈ R.
As the relation is reflexive, antisymmetric and transitive. Hence, it is a partial order relation.
G-1-SETS.pdf
SET FUNCTIONS
A function is sometimes called a map or mapping. It is a
correspondence, or relationship, between two sets called the
domain and range such that for each element of the domain
there corresponds exactly one element of the range.
In other words, a relation from A to B is a function F if
1. Every element of A is the first element of an ordered of F.
2. No two distinct ordered pairs in F have the same first element.
Kinds of Functions
1. One–One Function or Injective Function
The one-to-one function is also termed an injective
function. Here each element of the domain possesses
a different image or co-domain element for the
assigned function.
A function f: A → B is declared to be a one-one
function if different components in A have different
images or are associated with different elements in
B.
2. Onto Function or Surjective Function
A function f: A → B is declared to be an onto function
if each component in B has at least one pre-image in
A. i.e., If-Range of function f = Co-domain of function
f, then f is onto. The onto function is also termed a
subjective function.
3. Bijective Function or One One
and Onto Function
A function f: A → B is declared to be a
bijective function if it is both one-one and onto
function. In other words, we can say that
every element of set A is related to a different
element in set B, and there is not a single
element in set B that has been left out to be
connected to set A.
Any function f: A → B is said to be many-one if two
(or more than two) distinct components in A have
identical images in B. In a many-to-one function,
more than one element owns the same co-domain or
image. A function can be one to one or many to one
but not one to many.
4. Many-One Function
5. INTO FUNCTION
Any function f: A → B is said to be an into function if
there exists at least one element in B which does not
have a pre-image in A. This states that the elements
in set B are excess and are not equated to any
elements in set A.
Inverse of a Function
In mathematics a function, a, is said to be an inverse of another, b, if given the output of b a
returns the input value given to b.
The function f is called invertible, if its inverse function g exists.
Example:
Consider the functions a(x) = 5x + 2 and b(y) = (y-2)/5. Here function b is an inverse function of a.
When x is 1 the output of a is a(1) = 5(1) + 2 = 7. Using this output as y in function b gives b(7) =
(7-2)/5 = 1 which was the input value to function a.
Example 2:
Functions f(x)= x + 5 and g(x) = x − 5 are invertible since we use the value 1 to substitute x in the
first function and we get 6 as output. Then we use the output of the first function to substitute
the second function and we get 1 as output.
Composition of Functions
A composite function is a function whose input is another function. Two
functions f:A→B and g:B→C can be composed to give a composition gof. This
is a function from A to C defined by (gof)(x)=g(f(x))
Example
Let f(x) = x + 2 and g(x) = 2x + 1, find (fog)(x) and (gof)(x).
Solution
(fog)(x) = f(g(x)) = f(2x + 1) = x + 2 = 2x + 1 + 2 = 2x + 3
(gof)(x) = g(f(x)) = g(x+2) = 2x + 1 = 2(x + 2) +1 = 2x + 5
Hence, (fog)(x) ≠ (gof)(x)
Example 2:
Consider the functions A(x) = 5x + 2 and B(x) = x + 1.
Find (AoB)(x) and (BoA)(x).
AoB = A(B(x)) = 5(x+1) + 2
BoA = B(A(x)) = (5x + 2) + 1.
So AoB is not the same as BoA.
G-1-SETS.pdf
Question: A relation R on a non-empty set A is an equivalence relation if and only if it is
(a) Reflexive
(b) Symmetric and transitive
(c) Reflexive, symmetric and transitive
(d) None of these
Question: If A = {2, 4, 5}, B = {7, 8, 9}, then n(A x B) is equal to
(a) 6
(b) 9
(c) 3
(d) 0
Write the subsets of {1,2,3}.
If U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {1, 3, 5, 7, 9}. Find A′.
● Thank you for
listening!
65

More Related Content

Similar to G-1-SETS.pdf

Set theory- Introduction, symbols with its meaning
Set theory- Introduction, symbols with its meaningSet theory- Introduction, symbols with its meaning
Set theory- Introduction, symbols with its meaning
DipakMahurkar1
 
Sets Class XI Chapter 1
Sets Class XI Chapter 1Sets Class XI Chapter 1
Sets Class XI Chapter 1
Nishant Narvekar
 
2.1 Sets
2.1 Sets2.1 Sets
2.1 Sets
showslidedump
 
Discrete mathematics for diploma students
Discrete mathematics for diploma studentsDiscrete mathematics for diploma students
Discrete mathematics for diploma students
Zubair Khan
 
Chap2_SETS_PDF.pdf
Chap2_SETS_PDF.pdfChap2_SETS_PDF.pdf
Chap2_SETS_PDF.pdf
RatipornChomrit
 
SETS PPT-XI.pptx
SETS PPT-XI.pptxSETS PPT-XI.pptx
SETS PPT-XI.pptx
TamannaNayak5
 
SET AND ITS OPERATIONS
SET AND ITS OPERATIONSSET AND ITS OPERATIONS
SET AND ITS OPERATIONS
RohithV15
 
4898850.ppt
4898850.ppt4898850.ppt
4898850.ppt
UsamaManzoorLucky1
 
Set concepts
Set conceptsSet concepts
Set concepts
AarjavPinara
 
Sets class 11
Sets class 11Sets class 11
Sets class 11
Nitishkumar0105999
 
Set concepts
Set conceptsSet concepts
Sets and there different types.
Sets and there different types.Sets and there different types.
Sets and there different types.
Ashufb2323
 
Q1 Week 1 Lesson -Concepts of Sets and Operation on Sets.pptx
Q1 Week 1 Lesson -Concepts of Sets and Operation on Sets.pptxQ1 Week 1 Lesson -Concepts of Sets and Operation on Sets.pptx
Q1 Week 1 Lesson -Concepts of Sets and Operation on Sets.pptx
NovyFacun1
 
Explore the foundational concepts of sets in discrete mathematics
Explore the foundational concepts of sets in discrete mathematicsExplore the foundational concepts of sets in discrete mathematics
Explore the foundational concepts of sets in discrete mathematics
Dr Chetan Bawankar
 
mathematical sets.pdf
mathematical sets.pdfmathematical sets.pdf
mathematical sets.pdf
Jihudumie.Com
 
Lesson2_MathematicalLanguageAndSymbols _Lesson 2.1_VariablesAndTheLanguageOfS...
Lesson2_MathematicalLanguageAndSymbols _Lesson 2.1_VariablesAndTheLanguageOfS...Lesson2_MathematicalLanguageAndSymbols _Lesson 2.1_VariablesAndTheLanguageOfS...
Lesson2_MathematicalLanguageAndSymbols _Lesson 2.1_VariablesAndTheLanguageOfS...
LouelaDePaz
 
Sets
SetsSets
Types of sets
Types of setsTypes of sets
Types of sets
Mahitha Davala
 
Moazzzim Sir (25.07.23)CSE 1201, Week#3, Lecture#7.pptx
Moazzzim Sir (25.07.23)CSE 1201, Week#3, Lecture#7.pptxMoazzzim Sir (25.07.23)CSE 1201, Week#3, Lecture#7.pptx
Moazzzim Sir (25.07.23)CSE 1201, Week#3, Lecture#7.pptx
KhalidSyfullah6
 
Identifying subsets of a set
Identifying subsets of a setIdentifying subsets of a set
Identifying subsets of a set
MartinGeraldine
 

Similar to G-1-SETS.pdf (20)

Set theory- Introduction, symbols with its meaning
Set theory- Introduction, symbols with its meaningSet theory- Introduction, symbols with its meaning
Set theory- Introduction, symbols with its meaning
 
Sets Class XI Chapter 1
Sets Class XI Chapter 1Sets Class XI Chapter 1
Sets Class XI Chapter 1
 
2.1 Sets
2.1 Sets2.1 Sets
2.1 Sets
 
Discrete mathematics for diploma students
Discrete mathematics for diploma studentsDiscrete mathematics for diploma students
Discrete mathematics for diploma students
 
Chap2_SETS_PDF.pdf
Chap2_SETS_PDF.pdfChap2_SETS_PDF.pdf
Chap2_SETS_PDF.pdf
 
SETS PPT-XI.pptx
SETS PPT-XI.pptxSETS PPT-XI.pptx
SETS PPT-XI.pptx
 
SET AND ITS OPERATIONS
SET AND ITS OPERATIONSSET AND ITS OPERATIONS
SET AND ITS OPERATIONS
 
4898850.ppt
4898850.ppt4898850.ppt
4898850.ppt
 
Set concepts
Set conceptsSet concepts
Set concepts
 
Sets class 11
Sets class 11Sets class 11
Sets class 11
 
Set concepts
Set conceptsSet concepts
Set concepts
 
Sets and there different types.
Sets and there different types.Sets and there different types.
Sets and there different types.
 
Q1 Week 1 Lesson -Concepts of Sets and Operation on Sets.pptx
Q1 Week 1 Lesson -Concepts of Sets and Operation on Sets.pptxQ1 Week 1 Lesson -Concepts of Sets and Operation on Sets.pptx
Q1 Week 1 Lesson -Concepts of Sets and Operation on Sets.pptx
 
Explore the foundational concepts of sets in discrete mathematics
Explore the foundational concepts of sets in discrete mathematicsExplore the foundational concepts of sets in discrete mathematics
Explore the foundational concepts of sets in discrete mathematics
 
mathematical sets.pdf
mathematical sets.pdfmathematical sets.pdf
mathematical sets.pdf
 
Lesson2_MathematicalLanguageAndSymbols _Lesson 2.1_VariablesAndTheLanguageOfS...
Lesson2_MathematicalLanguageAndSymbols _Lesson 2.1_VariablesAndTheLanguageOfS...Lesson2_MathematicalLanguageAndSymbols _Lesson 2.1_VariablesAndTheLanguageOfS...
Lesson2_MathematicalLanguageAndSymbols _Lesson 2.1_VariablesAndTheLanguageOfS...
 
Sets
SetsSets
Sets
 
Types of sets
Types of setsTypes of sets
Types of sets
 
Moazzzim Sir (25.07.23)CSE 1201, Week#3, Lecture#7.pptx
Moazzzim Sir (25.07.23)CSE 1201, Week#3, Lecture#7.pptxMoazzzim Sir (25.07.23)CSE 1201, Week#3, Lecture#7.pptx
Moazzzim Sir (25.07.23)CSE 1201, Week#3, Lecture#7.pptx
 
Identifying subsets of a set
Identifying subsets of a setIdentifying subsets of a set
Identifying subsets of a set
 

Recently uploaded

Data Integration Basics: Merging & Joining Data
Data Integration Basics: Merging & Joining DataData Integration Basics: Merging & Joining Data
Data Integration Basics: Merging & Joining Data
Safe Software
 
Active Inference is a veryyyyyyyyyyyyyyyyyyyyyyyy
Active Inference is a veryyyyyyyyyyyyyyyyyyyyyyyyActive Inference is a veryyyyyyyyyyyyyyyyyyyyyyyy
Active Inference is a veryyyyyyyyyyyyyyyyyyyyyyyy
RaminGhanbari2
 
Calgary MuleSoft Meetup APM and IDP .pptx
Calgary MuleSoft Meetup APM and IDP .pptxCalgary MuleSoft Meetup APM and IDP .pptx
Calgary MuleSoft Meetup APM and IDP .pptx
ishalveerrandhawa1
 
"Mastering Graphic Design: Essential Tips and Tricks for Beginners and Profes...
"Mastering Graphic Design: Essential Tips and Tricks for Beginners and Profes..."Mastering Graphic Design: Essential Tips and Tricks for Beginners and Profes...
"Mastering Graphic Design: Essential Tips and Tricks for Beginners and Profes...
Anant Gupta
 
CiscoIconsLibrary cours de réseau VLAN.ppt
CiscoIconsLibrary cours de réseau VLAN.pptCiscoIconsLibrary cours de réseau VLAN.ppt
CiscoIconsLibrary cours de réseau VLAN.ppt
moinahousna
 
WPRiders Company Presentation Slide Deck
WPRiders Company Presentation Slide DeckWPRiders Company Presentation Slide Deck
WPRiders Company Presentation Slide Deck
Lidia A.
 
Litestack talk at Brighton 2024 (Unleashing the power of SQLite for Ruby apps)
Litestack talk at Brighton 2024 (Unleashing the power of SQLite for Ruby apps)Litestack talk at Brighton 2024 (Unleashing the power of SQLite for Ruby apps)
Litestack talk at Brighton 2024 (Unleashing the power of SQLite for Ruby apps)
Muhammad Ali
 
leewayhertz.com-AI agents for healthcare Applications benefits and implementa...
leewayhertz.com-AI agents for healthcare Applications benefits and implementa...leewayhertz.com-AI agents for healthcare Applications benefits and implementa...
leewayhertz.com-AI agents for healthcare Applications benefits and implementa...
alexjohnson7307
 
How Social Media Hackers Help You to See Your Wife's Message.pdf
How Social Media Hackers Help You to See Your Wife's Message.pdfHow Social Media Hackers Help You to See Your Wife's Message.pdf
How Social Media Hackers Help You to See Your Wife's Message.pdf
HackersList
 
WhatsApp Spy Online Trackers and Monitoring Apps
WhatsApp Spy Online Trackers and Monitoring AppsWhatsApp Spy Online Trackers and Monitoring Apps
WhatsApp Spy Online Trackers and Monitoring Apps
HackersList
 
July Patch Tuesday
July Patch TuesdayJuly Patch Tuesday
July Patch Tuesday
Ivanti
 
Evolution of iPaaS - simplify IT workloads to provide a unified view of data...
Evolution of iPaaS - simplify IT workloads to provide a unified view of  data...Evolution of iPaaS - simplify IT workloads to provide a unified view of  data...
Evolution of iPaaS - simplify IT workloads to provide a unified view of data...
Torry Harris
 
The Rise of AI in Cybersecurity How Machine Learning Will Shape Threat Detect...
The Rise of AI in Cybersecurity How Machine Learning Will Shape Threat Detect...The Rise of AI in Cybersecurity How Machine Learning Will Shape Threat Detect...
The Rise of AI in Cybersecurity How Machine Learning Will Shape Threat Detect...
digitalxplive
 
Best Practices for Effectively Running dbt in Airflow.pdf
Best Practices for Effectively Running dbt in Airflow.pdfBest Practices for Effectively Running dbt in Airflow.pdf
Best Practices for Effectively Running dbt in Airflow.pdf
Tatiana Al-Chueyr
 
High Profile Girls Call ServiCe Hyderabad 0000000000 Tanisha Best High Class ...
High Profile Girls Call ServiCe Hyderabad 0000000000 Tanisha Best High Class ...High Profile Girls Call ServiCe Hyderabad 0000000000 Tanisha Best High Class ...
High Profile Girls Call ServiCe Hyderabad 0000000000 Tanisha Best High Class ...
aslasdfmkhan4750
 
Recent Advancements in the NIST-JARVIS Infrastructure
Recent Advancements in the NIST-JARVIS InfrastructureRecent Advancements in the NIST-JARVIS Infrastructure
Recent Advancements in the NIST-JARVIS Infrastructure
KAMAL CHOUDHARY
 
Implementations of Fused Deposition Modeling in real world
Implementations of Fused Deposition Modeling  in real worldImplementations of Fused Deposition Modeling  in real world
Implementations of Fused Deposition Modeling in real world
Emerging Tech
 
How to Build a Profitable IoT Product.pptx
How to Build a Profitable IoT Product.pptxHow to Build a Profitable IoT Product.pptx
How to Build a Profitable IoT Product.pptx
Adam Dunkels
 
Using LLM Agents with Llama 3, LangGraph and Milvus
Using LLM Agents with Llama 3, LangGraph and MilvusUsing LLM Agents with Llama 3, LangGraph and Milvus
Using LLM Agents with Llama 3, LangGraph and Milvus
Zilliz
 
Google I/O Extended Harare Merged Slides
Google I/O Extended Harare Merged SlidesGoogle I/O Extended Harare Merged Slides
Google I/O Extended Harare Merged Slides
Google Developer Group - Harare
 

Recently uploaded (20)

Data Integration Basics: Merging & Joining Data
Data Integration Basics: Merging & Joining DataData Integration Basics: Merging & Joining Data
Data Integration Basics: Merging & Joining Data
 
Active Inference is a veryyyyyyyyyyyyyyyyyyyyyyyy
Active Inference is a veryyyyyyyyyyyyyyyyyyyyyyyyActive Inference is a veryyyyyyyyyyyyyyyyyyyyyyyy
Active Inference is a veryyyyyyyyyyyyyyyyyyyyyyyy
 
Calgary MuleSoft Meetup APM and IDP .pptx
Calgary MuleSoft Meetup APM and IDP .pptxCalgary MuleSoft Meetup APM and IDP .pptx
Calgary MuleSoft Meetup APM and IDP .pptx
 
"Mastering Graphic Design: Essential Tips and Tricks for Beginners and Profes...
"Mastering Graphic Design: Essential Tips and Tricks for Beginners and Profes..."Mastering Graphic Design: Essential Tips and Tricks for Beginners and Profes...
"Mastering Graphic Design: Essential Tips and Tricks for Beginners and Profes...
 
CiscoIconsLibrary cours de réseau VLAN.ppt
CiscoIconsLibrary cours de réseau VLAN.pptCiscoIconsLibrary cours de réseau VLAN.ppt
CiscoIconsLibrary cours de réseau VLAN.ppt
 
WPRiders Company Presentation Slide Deck
WPRiders Company Presentation Slide DeckWPRiders Company Presentation Slide Deck
WPRiders Company Presentation Slide Deck
 
Litestack talk at Brighton 2024 (Unleashing the power of SQLite for Ruby apps)
Litestack talk at Brighton 2024 (Unleashing the power of SQLite for Ruby apps)Litestack talk at Brighton 2024 (Unleashing the power of SQLite for Ruby apps)
Litestack talk at Brighton 2024 (Unleashing the power of SQLite for Ruby apps)
 
leewayhertz.com-AI agents for healthcare Applications benefits and implementa...
leewayhertz.com-AI agents for healthcare Applications benefits and implementa...leewayhertz.com-AI agents for healthcare Applications benefits and implementa...
leewayhertz.com-AI agents for healthcare Applications benefits and implementa...
 
How Social Media Hackers Help You to See Your Wife's Message.pdf
How Social Media Hackers Help You to See Your Wife's Message.pdfHow Social Media Hackers Help You to See Your Wife's Message.pdf
How Social Media Hackers Help You to See Your Wife's Message.pdf
 
WhatsApp Spy Online Trackers and Monitoring Apps
WhatsApp Spy Online Trackers and Monitoring AppsWhatsApp Spy Online Trackers and Monitoring Apps
WhatsApp Spy Online Trackers and Monitoring Apps
 
July Patch Tuesday
July Patch TuesdayJuly Patch Tuesday
July Patch Tuesday
 
Evolution of iPaaS - simplify IT workloads to provide a unified view of data...
Evolution of iPaaS - simplify IT workloads to provide a unified view of  data...Evolution of iPaaS - simplify IT workloads to provide a unified view of  data...
Evolution of iPaaS - simplify IT workloads to provide a unified view of data...
 
The Rise of AI in Cybersecurity How Machine Learning Will Shape Threat Detect...
The Rise of AI in Cybersecurity How Machine Learning Will Shape Threat Detect...The Rise of AI in Cybersecurity How Machine Learning Will Shape Threat Detect...
The Rise of AI in Cybersecurity How Machine Learning Will Shape Threat Detect...
 
Best Practices for Effectively Running dbt in Airflow.pdf
Best Practices for Effectively Running dbt in Airflow.pdfBest Practices for Effectively Running dbt in Airflow.pdf
Best Practices for Effectively Running dbt in Airflow.pdf
 
High Profile Girls Call ServiCe Hyderabad 0000000000 Tanisha Best High Class ...
High Profile Girls Call ServiCe Hyderabad 0000000000 Tanisha Best High Class ...High Profile Girls Call ServiCe Hyderabad 0000000000 Tanisha Best High Class ...
High Profile Girls Call ServiCe Hyderabad 0000000000 Tanisha Best High Class ...
 
Recent Advancements in the NIST-JARVIS Infrastructure
Recent Advancements in the NIST-JARVIS InfrastructureRecent Advancements in the NIST-JARVIS Infrastructure
Recent Advancements in the NIST-JARVIS Infrastructure
 
Implementations of Fused Deposition Modeling in real world
Implementations of Fused Deposition Modeling  in real worldImplementations of Fused Deposition Modeling  in real world
Implementations of Fused Deposition Modeling in real world
 
How to Build a Profitable IoT Product.pptx
How to Build a Profitable IoT Product.pptxHow to Build a Profitable IoT Product.pptx
How to Build a Profitable IoT Product.pptx
 
Using LLM Agents with Llama 3, LangGraph and Milvus
Using LLM Agents with Llama 3, LangGraph and MilvusUsing LLM Agents with Llama 3, LangGraph and Milvus
Using LLM Agents with Llama 3, LangGraph and Milvus
 
Google I/O Extended Harare Merged Slides
Google I/O Extended Harare Merged SlidesGoogle I/O Extended Harare Merged Slides
Google I/O Extended Harare Merged Slides
 

G-1-SETS.pdf

  • 1. S E T S A set is an unordered collection of different elements. A set can be written explicitly by listing its elements using set bracket. If the order of the elements is changed or any element of a set is repeated, it does not make any changes in the set.
  • 2. Some Example of Sets ● A set of all positive integers ● A set of all the planets in the solar system ● A set of all the states in India ● A set of all the lowercase letters of the alphabet
  • 3. Cardinality of a Set ● Cardinality of a set S, denoted by |S|, is the number of elements of the set. The number is also referred as the cardinal number. If a set has an infinite number of elements, its cardinality is ∞. Example − |{1,4,3,5}|=4,|{1,2,3,4,5,…}|=∞
  • 4. TYPES OF SETS Sets can be classified into many types. Some of which are finite, infinite, subset, universal, proper, singleton set, etc.
  • 5. Finite Set ● A set which consists of a definite number of elements is called a finite set. Example: A set of natural numbers up to 10. A = {1,2,3,4,5,6,7,8,9,10}
  • 6. Empty Set ● A set which does not contain any element is called an empty set or void set or null set. It is denoted by { } or Ø. Example ● A set of apples in the basket of grapes is an example of an empty set because in a grapes basket there are no apples present.
  • 7. Singleton Set ● A set which contains a single element is called a singleton set. Example: There is only one apple in the basket. Example 2: A = {10}
  • 8. Infinite Set ● A set which is not finite is called an infinite set. We use ellipsis to indicate an infinite set. ● Example: A set of all natural numbers. ● A = {1,2,3,4,5,6,7,8,9……}
  • 9. Equivalent set ● If the number of elements is the same for two different sets, then they are called equivalent sets. The order of sets does not matter here. It is represented as: n(A) = n(B) where A and B are two different sets with the same number of elements. Example: If A = {1,2,3,4} and B = {Red, Blue, Green, Black} In set A, there are four elements and in set B also there are four elements. Therefore, set A and set B are equivalent.
  • 10. Equal sets ● The two sets A and B are said to be equal if they have exactly the same elements, the order of elements do not matter. Example: A = {1,2,3,4} and B = {4,3,2,1} A = B
  • 11. Disjoint Sets ● The two sets A and B are said to be disjoint if the set does not contain any common element. Example: Set A = {1,2,3,4} and set B = {5,6,7,8} are disjoint sets, because there is no common element between them.
  • 12. Subsets ● A set ‘A’ is said to be a subset of B if every element of A is also an element of B, denoted as A ⊆ B. Even the null set is considered to be the subset of another set. In general, a subset is a part of another set. Example: A = {1,2,3} Then {1,2} ⊆ A. Similarly, other subsets of set A are: {1},{2},{3},{1,2},{2,3},{1,3},{1,2,3},{}. Note: The set is also a subset of itself. If A is not a subset of B, then it is denoted as A⊄B.
  • 13. Proper Subset ● If A ⊆ B and A ≠ B, then A is called the proper subset of B and it can be written as A⊂B. Example: If A = {2,5,7} is a subset of B = {2,5,7} then it is not a proper subset of B = {2,5,7} But, A = {2,5} is a subset of B = {2,5,7} and is a proper subset also.
  • 14. Superset A superset can be defined as a set of elements containing all of the elements of another set. Set A is said to be the superset of B if all the elements of set B are the elements of set A. It is represented as A ⊃ B. For example, if set A = {1, 2, 3, 4} and set B = {1, 3, 4}, then set A is the superset of B.
  • 15. Universal Set ● A set which contains all the sets relevant to a certain condition is called the universal set. It is the set of all possible values. Example: If A = {1,2,3} and B {2,3,4,5}, then universal set here will be: U = {1,2,3,4,5}
  • 16. SET OPERATIONS 1. Intersection (denoted by ∩): The intersection of two sets A and B is the set of all elements that are in both A and B. For example: if A = {1, 3, 8} and B = {-9, 22, 3}, then A ∩ B = {3}1.
  • 17. 2. Disjoint Sets ● Two sets are disjoint if they have no elements in common1. In other words, A and B are disjoint if their intersection is the empty set (∅)1.
  • 18. 3. Union (denoted by ∪) ● The union of two sets A and B is the set of all elements that are in A or in B or in both1. For example, if A = {2, 5, 8} and B = {7, 5, 22}, then A ∪ B = {2, 5, 8, 7, 22}1.
  • 19. 4. Complement of a Set ● The complement of a set A (denoted by A′) is the set of elements which are not in set A.
  • 20. 5. Set Difference/ Relative Complement ● The set difference of sets A and B (denoted by A–B) is the set of elements which are only in A but not in B. Example − If A={10,11,12,13} and B={13,14,15}, then (A−B)={10,11,12} and (B−A)={14,15}. Here, we can see (A−B)≠(B−A)
  • 21. Power Set ● Power set of a set S is the set of all subsets of S including the empty set. The cardinality of a power set of a set S of cardinality n is 2n. Power set is denoted as P(S). Example − For a set S={a,b,c,d} let us calculate the subsets − Subsets with 0 elements − {∅} (the empty set) Subsets with 1 element − {a},{b},{c},{d} Subsets with 2 elements − {a,b},{a,c},{a,d},{b,c},{b,d},{c,d} Subsets with 3 elements − {a,b,c},{a,b,d},{a,c,d},{b,c,d} Subsets with 4 elements − {a,b,c,d}
  • 22. ● Hence, P(S)= ● {{∅},{a},{b},{c},{d},{a,b},{a,c},{a,d},{b,c},{b,d},{c,d},{a,b,c},{a,b,d},{a,c,d},{b,c,d},{a,b,c,d}} ● |P(S)|=24=16 ● Note − The power set of an empty set is also an empty set. ● |P({∅})|=20=1
  • 24. WHAT IS SET RELATIONS?  A set relation is a fundamental concept in mathematics that allows us to describe and understand the connections or associations between elements within sets. In the context of set theory, a relation is essentially a set of ordered pairs.  Set relations play a crucial role in various fields, including mathematics, computer science, data analysis, and beyond. They provide a foundation for understanding relationships, making comparisons, and solving problems. Significance: ● Understanding Relationships ● Comparisons and Classification ● Problem solving ● Foundation for further concepts
  • 25.  A set of ordered pairs is defined as a relation.’  This mapping depicts a relation from set A into set B. A relation from A to B is a subset of A x B. The ordered pairs are (1,c),(2,n),(5,a),(7,n). For defining a relation, we use the notation where, • set {1, 2, 5, 7} represents the domain. • set {a, c, n} represents the range.
  • 26. Finding the domain, range, and codomain of a relation using the roster method involves identifying the elements that belong to each of these sets based on the ordered pairs in the relation. Let's break down how to find each of these components using an example.
  • 27. Example Relation: Consider the relation R between set A = {1, 2, 3} and set B = {4, 5, 6} defined by the following ordered pairs: R = {(1, 4), (2, 5), (3, 6)} Domain: The domain of a relation consists of all the first elements (the elements from the left side of the ordered pairs) in the relation. In this case, the domain of R is the set of all first elements of the ordered pairs in R. Domain(R) = {1, 2, 3} Range: The range of a relation consists of all the second elements (the elements from the right side of the ordered pairs) in the relation. In this case, the range of R is the set of all second elements of the ordered pairs in R. Range(R) = {4, 5, 6}
  • 28. Codomain: The codomain is the set that specifies the possible values for the second elements of the ordered pairs in the relation. It is predetermined and doesn't depend on the actual ordered pairs in the relation. In this case, the codomain of R is set B, which is {4, 5, 6}. So, for the given relation R using the roster method: Domain(R) = {1, 2, 3,} Range(R) = {4, 5, 6} Codomain(R) = {4, 5, 6}
  • 29. 9 8 TYPES OF SET RELATIONS Empty Relation 1 2 3 4 6 7 Full Relation Identity Relation Inverse Relation Symmetric Relation Transitive Relation Equivalence Relation Partial Order Relation 5 Reflexive Relation 10 Anti-Symmetric Relation
  • 30.  An empty relation (or void relation) is one in which there is no relation between any elements of a set. It is one of the simplest types of set relations. For empty relation, R = φ ⊂ A × A  In this case, there are no ordered pairs in the relation R, which means there is no connection or relationship between any element. EMPTY RELATION • Set X: {1, 3, 5} • Set Y: {2, 4, 6} R={ }; This is a clear example of an empty relation, signifying the absence of any association or interaction between the two sets.
  • 31. FULL RELATIONS A Full relation is a type of relation in which every element of a set is related to each other, also known as the universal relation or complete relation, which is the opposite of an empty relation. It represents a set relation where every possible pair of elements from two sets is included. The full relation is often denoted as U (for universal) or sometimes as the set of all possible pairs of elements, denoted as U = {(a, b) | a ∈ Set A, b ∈ Set B}. Example: Consider two sets: ● Set A: {Alice, Bob, Carol} ● Set B: {X, Y} The full relation between these sets would be represented as a set of ordered pairs: R = {(Alice, X), (Alice, Y), (Bob, X), (Bob, Y), (Carol, X), (Carol, Y)} In this relation R, every possible pair of elements from Set A and Set B is included, resulting in a complete and exhaustive set of connections.
  • 32. Imagine a set of ordered pairs that connects every element in Set A to every element in Set B, forming a complete web of connections. The formula for a full relation R between two sets A and B is typically represented as: R = A × B • Here, "×" represents the Cartesian product of sets A and B, which generates all possible ordered pairs of elements from A and B. Therefore, R includes all the ordered pairs, making it the full relation between A and B. Example 2: Consider two sets: ● Set A: {1,4, 8l} ● Set B: {x, y} The full relation between these sets would be represented as a set of ordered pairs: R = {(1, X), (1, Y), (4, X), (4, Y), (8, X), (8, Y)} In this relation R, every possible pair of elements from Set A and Set B.
  • 33. IDENTITY RELATIONS  Set A is a relation where every element of A is related to itself only. This means that if you have a set “A”, the identity relation “I” contains pairs of elements from “A” where both elements in each pair are the same. This relation reflects the concept of self-identity or equality within the set. The condition for the identity relation on set A is represented as: I = {(a, a) | a ∈ A} In this condition: (a, a) represents an ordered pair where an element "a" is related to itself. "a ∈ A" indicates that "a" is an element of set A. Example: set A={a, b, c}, the identity relation will be I = {a, a},{b, b}, {c, c} For Identity relation I = {(a, a), a ∈ A}
  • 34. Identity relation consists of all such ordered pairs where each element in set A is related only to itself. Example: Consider a set A that represents the set of natural numbers less than or equal to 5: A = {1, 2, 3, 4, 5} The identity relation (Id_A) on this set A would consist of ordered pairs where each element is related to itself. Here's what the identity relation would look like for this set: Id_A = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)} In this example, every element from the set A is related only to itself, which is the defining characteristic of the identity relation.
  • 35. INVERSE RELATION Inverse relation is seen when a set has elements which are inverse pair of another set. If (x, y) ∈ R, then (y, x) ∈ R^(-1) and vice versa. i.e., If R is from A to B, then R^(-1) is from B to A. Thus, if R is a subset of A x B, then R-1 is a subset of B x A. An inverse relation is the inverse of a relation and is obtained by interchanging the elements of each ordered pair of the given relation.
  • 36. Example: Have a look at the following relations and their inverse relations on two sets A = {a, b, c, d, e} and B = {1, 2, 3, 4, 5}. • If R = {(a, 2), (b, 4), (c, 1)} ⇔ R-1 = {(2, a), (4, b), (1, c)} • If R = {(c, 1), (b, 2), (a, 3)} ⇔ R-1 = {(1, c), (2, b), (3, a)} • If R = {(b, 3), (c, 2), (e, 1)} ⇔ R-1 = {(3, b), (2, c), (1, e)} The formula for the inverse relation R^(-1) of a relation R is defined as: R^(-1) = {(b, a) | (a, b) ∈ R} In this formula: (a, b) represents an ordered pair in the original relation R. (b, a) represents the corresponding ordered pair in the inverse relation R^(-1). So, R^(-1) contains all the ordered pairs from R, but with their elements reversed in order.
  • 37. A relation R on a set A is said to be reflexive if, for every element a in set A, (a, a) belongs to R. In other words, every element is related to itself. Reflexive relations often represent properties of elements that are inherent to the elements themselves. They are symbolically represented as (a, a) ∈ R for all a in A. REFLEXIVE RELATIONS Where a is the element, A is the set and R is the relation.
  • 38. Example: (a, a) ∈ R A = {1, 2, 3} R1 = {(1, 1), (1, 2)} not Reflexive R2 = {(1, 1), (1, 2)} Reflexive R3 = {(1, 1), (2, 2), (3, 3)} Reflexive RA = {(1, 1), (2, 2), (3, 2)} not Reflexive R5 = { } not Reflexive R6 = A x A Reflexive How to calculate how many Reflexive relations are there if we have an n element. N= Example: A= {1,2,3} N = 2(9-3) N = 26 N = 64
  • 39. A symmetric relation is a fundamental concept in discrete mathematics that describes a specific type of binary relation between elements in a set. It possesses a key property known as symmetry, which means that if an element 'a' is related to an element 'b,' then 'b' is also related to ‘a.’ Condition for symmetric relation: (a,b) ∈ R ⇒ (b,a) ∈ R for all a, b ∈ A. aRb ⇒ bRa for all a,b ∈ A. SYMMETRIC RELATION
  • 40. Example: Consider a set of cities and a relation R defined on pairs of cities. We define R as follows: (City A, City B) is in relation R if and only if there is a direct flight between City A and City B. Demonstrating Symmetry: • Suppose there is a direct flight from City A to City B. According to the definition, (City A, City B) ∈ R. • Now, since there is a direct flight, it also implies that there is a direct flight from City B to City A. • Therefore, (City B, City A) must also be in relation R. This example illustrates that if there's a direct flight from A to B, there's also a direct flight from B to A, satisfying the symmetry property. Symmetric relations often appear in real-world scenarios, such as transportation networks. In this example, the symmetric relation R reflects the bi-directional nature of direct flights between cities.
  • 41. Symmetric Relation Formula: The number of symmetric relations on a set with ‘n’ elements is given by the formula: N = Example: A={1,2,3} N= 2n(n+1)/2 N = 2(3)(3+1)/2 = 2(3)(4)/2 = 212/2 = 26 = 64 Example 1: Suppose R is a relation on a set P where A = {3, 4, 5} and R = {(3,3), (3,4), (3,5), (4,5), (5,3)}. Check if R is a symmetric relation. Solution: As we can view that (3,4) ∈ R. For R to be symmetric (4, 3) should be in R although (4, 3) ∉ R.Also (4,5)∈ R but (5, 4) ∉ R Therefore, R is not a symmetric relation. Example 2: Let Z be the set of two female kids (z, x) in a family and R be a relation. Solution: Let z, x ∈ Z. If “z” is the sister of “x”, then “x” has to be the sister of “z”. We can say that, R = {(z, x), (x, z)} So, R is symmetric.
  • 42.  A relation R on a set A is said to be asymmetric if and only if (a,b)∈R , then (b,a)∉R , for all a,b∈A. In other words, an asymmetric relation is the opposite of a symmetric relation. Example: The relation R “is a parent of a and b” is asymmetric since if a is the parent of b , then b cannot be the parent of a  A relation R on a set A is known as asymmetric relation if no (b,a) ∈ R when (a,b) ∈ R or we can even say that relation R on set A is symmetric if only if (a,b) ∈ R⟹(b, a) ∉R. ASYMMETRIC RELATION
  • 43. Anti-symmetric Relation: A relation R on a set A is said to be antisymmetric, if aRb and bRa holds if and only if when a=b. In other words, (a,b)∉R and (b,a)∉R if a≠b. Example: Let us consider A to be the set on which the relation R is defined, then R is said to be antisymmetric when aRb and bRa⇒a=b where a, b∈A i.e. If (a, b)∈R & (b, a)∈R, then a=b. where, a∈A and b∈B. ANTI-SYMMETRIC RELATION
  • 44.  Transitive relations are binary relations in set theory that are defined on a set A such that if a is related to b and b is related to c, then element a must be related to element c, for a, b, c in set A.  A binary relation R defined on a set A is said to be a transitive relation for all a, b, c in A if a R b and b R c, then a R c, that is, if a is related to b and b is related to c, then a must be related to c. Mathematically, we can write it as: a relation R defined on a set A is a transitive relation for all a, b, c ∈ A, if (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R. TRANSITIVE RELATION
  • 45.  Example 1:  Define a relation R on a set A = {a, b, c} as R = {(a, b), (b, c), (b, b)}. Determine if R is a transitive relation.  Solution: As we can see that (a, b) ∈ R and (b, c) ∈ R, and for R to be transitive (a, c) ∈ R must hold, but (a, c) ∉ R. So, R is not a transitive relation.  Answer: R is not a transitive relation  Example 2: Consider A ={1, 2, 3, 4} R1={(1,1), (1,2), (2,3), (1,3), (4,4) Transitive R2={(1,1), (1,2), (2,1), (2,2), (3,3), (4,4) Transitive R3={(1,3), (2,1)} Not Transitive
  • 46. EQUIVALENCE RELATION  An equivalence relation is a binary relation defined on a set X such that the relation is reflexive, symmetric and transitive. If any of the three conditions (reflexive, symmetric and transitive) does not hold, the relation cannot be an equivalence relation. The equivalence relation divides the set into disjoint equivalence classes. Any two elements of the set are said to be equivalent if and only if they belong to the same equivalence class.  A relation R defined in a set is called an Equivalence relation if it satisfy the following:
  • 47. Example2: Consider a group of friends who are trying to determine if they have similar tastes in music. They decide to categorize their music preferences based on whether they have the same favorite genre. Defining the Relation: Let's represent each friend by a letter: A, B, C, D, and so on. We say that two friends, denoted as (a, b) are in the same group if they share the same favorite music genre. Demonstrating Equivalence: • Reflexivity: Each person, like A, has themselves as their friend (a, a). So, everyone is in their own group, indicating that they share the same taste as themselves. • Symmetry: If A and B are in the same group, it means they have the same favorite genre. Therefore, B and A are also in the same group, showing that the relationship is mutual. • Transitivity: If A and B are in the same group (same music taste), and B and C are also in the same group, then it follows that A and C share the same favorite genre. So, the transitivity property holds. • Equivalence Classes: In this example, each group consists of friends who share the same favorite music genre. For instance, there may be a group of friends who all love rock music, another group who adore jazz, and so on.
  • 48.  Also known as partially ordered sets or posets, are a fundamental concept in mathematics and computer science. They are a specific type of binary relation that satisfies three key properties: • Reflexivity: For every element 'a' in the set, 'a' is related to itself. This is represented as: a ≤ a. • Antisymmetry: If 'a' is related to 'b' and 'b' is related to 'a', then 'a' and 'b' must be the same element. In other words, if a ≤ b and b ≤ a, then a = b. • Transitivity: If 'a' is related to 'b' and 'b' is related to 'c', then 'a' is related to 'c'. This is represented as: If a ≤ b and b ≤ c, then a ≤ c. PARTIAL ORDER RELATIONS  Symbols • Partial Order Relations are typically denoted using the symbol '≤' (less than or equal to) or other similar symbols like '⊆' (subset) or '⊇' (superset) depending on the context. For instance: a ≤ b denotes that element 'a' is related to element 'b'. A ⊆ B represents that set 'A' is a subset of set 'B'.
  • 49. Example 1: Consider the set of natural numbers (N) and the relation '≤' defined on N, where 'a ≤ b' if and only if 'a' is divisible by 'b' without a remainder. • Reflexivity: For any natural number 'a', 'a' is divisible by itself without a remainder, so 'a ≤ a' holds. • Antisymmetry: If 'a ≤ b' and 'b ≤ a', then 'a' and 'b' must be the same number. For instance, if '4 ≤ 2' (since 4 is divisible by 2) and '2 ≤ 4' (since 2 is divisible by 4), then 'a = b = 2’. • Transitivity: If 'a ≤ b' and 'b ≤ c', then 'a ≤ c'. For example, if '8 ≤ 4' (since 8 is divisible by 4) and '4 ≤ 2' (since 4 is divisible by 2), then '8 ≤ 2' (since 8 is divisible by 2). This example illustrates how the relation '≤' on natural numbers satisfies the properties of a partial order relation, making it a partial order relation on the set of natural numbers.
  • 50. Example 2:  Show whether the relation (x, y) ∈ R, if, x ≥ y defined on the set of integers is a partial order relation. Consider the set A = {1, 2, 3, 4} containing four integers. Find the relation for this set such as R = {(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3), (1, 1), (2, 2), (3, 3), (4, 4)}. • Reflexive: The relation is reflexive as for every a ∈ A. (a, a) ∈ R, i.e. (1, 1), (2, 2), (3, 3), (4, 4) ∈ R. • Antisymmetric: The relation is antisymmetric as whenever (a, b) and (b, a) ∈ R, we have a = b. • Transitive: The relation is transitive as whenever (a, b) and (b, c) ∈ R, we have (a, c) ∈ R. Example: (4, 2) ∈ R and (2, 1) ∈ R, implies (4, 1) ∈ R. As the relation is reflexive, antisymmetric and transitive. Hence, it is a partial order relation.
  • 53. A function is sometimes called a map or mapping. It is a correspondence, or relationship, between two sets called the domain and range such that for each element of the domain there corresponds exactly one element of the range. In other words, a relation from A to B is a function F if 1. Every element of A is the first element of an ordered of F. 2. No two distinct ordered pairs in F have the same first element.
  • 54. Kinds of Functions 1. One–One Function or Injective Function The one-to-one function is also termed an injective function. Here each element of the domain possesses a different image or co-domain element for the assigned function. A function f: A → B is declared to be a one-one function if different components in A have different images or are associated with different elements in B.
  • 55. 2. Onto Function or Surjective Function A function f: A → B is declared to be an onto function if each component in B has at least one pre-image in A. i.e., If-Range of function f = Co-domain of function f, then f is onto. The onto function is also termed a subjective function.
  • 56. 3. Bijective Function or One One and Onto Function A function f: A → B is declared to be a bijective function if it is both one-one and onto function. In other words, we can say that every element of set A is related to a different element in set B, and there is not a single element in set B that has been left out to be connected to set A.
  • 57. Any function f: A → B is said to be many-one if two (or more than two) distinct components in A have identical images in B. In a many-to-one function, more than one element owns the same co-domain or image. A function can be one to one or many to one but not one to many. 4. Many-One Function
  • 58. 5. INTO FUNCTION Any function f: A → B is said to be an into function if there exists at least one element in B which does not have a pre-image in A. This states that the elements in set B are excess and are not equated to any elements in set A.
  • 59. Inverse of a Function In mathematics a function, a, is said to be an inverse of another, b, if given the output of b a returns the input value given to b. The function f is called invertible, if its inverse function g exists. Example: Consider the functions a(x) = 5x + 2 and b(y) = (y-2)/5. Here function b is an inverse function of a. When x is 1 the output of a is a(1) = 5(1) + 2 = 7. Using this output as y in function b gives b(7) = (7-2)/5 = 1 which was the input value to function a. Example 2: Functions f(x)= x + 5 and g(x) = x − 5 are invertible since we use the value 1 to substitute x in the first function and we get 6 as output. Then we use the output of the first function to substitute the second function and we get 1 as output.
  • 60. Composition of Functions A composite function is a function whose input is another function. Two functions f:A→B and g:B→C can be composed to give a composition gof. This is a function from A to C defined by (gof)(x)=g(f(x)) Example Let f(x) = x + 2 and g(x) = 2x + 1, find (fog)(x) and (gof)(x). Solution (fog)(x) = f(g(x)) = f(2x + 1) = x + 2 = 2x + 1 + 2 = 2x + 3 (gof)(x) = g(f(x)) = g(x+2) = 2x + 1 = 2(x + 2) +1 = 2x + 5 Hence, (fog)(x) ≠ (gof)(x)
  • 61. Example 2: Consider the functions A(x) = 5x + 2 and B(x) = x + 1. Find (AoB)(x) and (BoA)(x). AoB = A(B(x)) = 5(x+1) + 2 BoA = B(A(x)) = (5x + 2) + 1. So AoB is not the same as BoA.
  • 63. Question: A relation R on a non-empty set A is an equivalence relation if and only if it is (a) Reflexive (b) Symmetric and transitive (c) Reflexive, symmetric and transitive (d) None of these Question: If A = {2, 4, 5}, B = {7, 8, 9}, then n(A x B) is equal to (a) 6 (b) 9 (c) 3 (d) 0
  • 64. Write the subsets of {1,2,3}. If U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {1, 3, 5, 7, 9}. Find A′.
  • 65. ● Thank you for listening! 65