- 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
- 23. SET RELATIONS
- 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.
- 52. SET FUNCTIONS
- 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