3. SET OPERATIONS
Set operations is a concept similar to fundamental operations on
numbers.
Sets in math deal with a finite collection of objects, be it numbers,
alphabets, or any real-world objects.
Sometimes a necessity arises wherein we need to establish the
relationship between two or more sets.
There comes the concept of set operations.
There are four main set operations which include set union, set
intersection, set complement, and set difference.
4. THERE ARE FOUR MAIN KINDS OF
SET OPERATIONS WHICH ARE:
Union of sets
Intersection of sets
Complement of a set
Difference between sets/Relative Complement
5. UNION OF SETS
For two given sets A and B,
A∪B (read as A union B) is the set of distinct elements that belong to
set A and B or both.
The number of elements in A ∪ B is given by
n(A∪B) = n(A) + n(B) − n(A∩B),
where n(X) is the number of elements in set X.
To understand this set operation of the union of sets better, let us
consider an example:
If A = {1, 2, 3, 4} and B = {4, 5, 6, 7}, then the union of A and B is
given by A ∪ B = {1, 2, 3, 4, 5, 6, 7}.
6. INTERSECTION OF SETS
For two given sets A and B, A∩B (read as A intersection B) is the set of
common elements that belong to set A and B.
The number of elements in A∩B is given by
n(A∩B) = n(A)+n(B)−n(A∪B),
where n(X) is the number of elements in set X.
To understand this set operation of the intersection of sets better, let
us consider an example:
If A = {1, 2, 3, 4} and B = {3, 4, 5, 7}, then the intersection of A and B
is given by A ∩ B = {3, 4}.
7. SET DIFFERENCE
The set operation difference between sets implies subtracting the
elements from a set which is similar to the concept of the difference
between numbers.
The difference between sets A and B denoted as A − B lists all the
elements that are in set A but not in set B.
To understand this set operation of set difference better, let us
consider an example: If
A = {1, 2, 3, 4} and B = {3, 4, 5, 7}, then the difference between sets
A and B is given by A - B = {1, 2}.
8. COMPLEMENT OF SETS
The complement of a set A denoted as A′ or Ac (read as A
complement) is defined as the set of all the elements in the given
universal set(U) that are not present in set A.
To understand this set operation of complement of sets better, let us
consider an example:
If U = {1, 2, 3, 4, 5, 6, 7, 8, 9} and A = {1, 2, 3, 4}, then the
complement of set A is given by A' = {5, 6, 7, 8, 9}.
9.
10. TYPES OF RELATIONS
There are 8 main types of relations which include:
Empty Relation
Universal Relation
Identity Relation
Inverse Relation
Reflexive Relation
Symmetric Relation
Transitive Relation
Equivalence Relation
11. TYPES OF RELATIONS(CONT….)
Empty Relation
An empty relation (or void relation) is one in which there is no relation
between any elements of a set. For example, if set A = {1, 2, 3} then, one of
the void relations can be R = {x, y} where, |x – y| = 8. For empty relation,
R = φ ⊂ A × A
Universal Relation
A universal (or full relation) is a type of relation in which every element of a
set is related to each other. Consider set A = {a, b, c}. Now one of the
universal relations will be R = {x, y} where, |x – y| ≥ 0. For universal relation,
R = A × A
12. TYPES OF RELATIONS(CONT….)
Identity Relation
In an identity relation, every element of a set is related to itself only.
For example, in a 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}
Inverse Relation
Inverse relation is seen when a set has elements which are inverse
pairs of another set. For example if set A = {(a, b), (c, d)}, then inverse
relation will be R-1 = {(b, a), (d, c)}. So, for an inverse relation,
R-1 = {(b, a): (a, b) ∈ R}
13. TYPES OF RELATIONS(CONT….)
Reflexive Relation
In a reflexive relation, every element maps to itself. For example, consider a set
A = {1, 2,}. Now an example of reflexive relation will be R = {(1, 1), (2, 2), (1, 2),
(2, 1)}. The reflexive relation is given by-
(a, a) ∈ R
Symmetric Relation
In a symmetric relation, if a=b is true then b=a is also true. In other words, a
relation R is symmetric only if (b, a) ∈ R is true when (a,b) ∈ R. An example of
symmetric relation will be R = {(1, 2), (2, 1)} for a set A = {1, 2}. So, for a
symmetric relation,
aRb ⇒ bRa, ∀ a, b ∈ A
.
14. TYPES OF RELATIONS(CONT….)
Transitive Relation
For transitive relation, if (x, y) ∈ R, (y, z) ∈ R, then (x, z) ∈ R. For a
transitive relation,
aRb and bRc ⇒ aRc ∀ a, b, c ∈ A
Equivalence Relation
If a relation is reflexive, symmetric and transitive at the same time it
is known as an equivalence relation
15. PARTIALLY ORDERING
Partial Order Relations
A relation R on a set A is called a partial order relation if it satisfies
the following three properties:
Relation R is Reflexive, i.e. aRa ∀ a∈A.
Relation R is Antisymmetric, i.e., aRb and bRa ⟹ a = b.
Relation R is transitive, i.e., aRb and bRc ⟹ aRc.
