Sets and relations

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1. Sets and relations BY:SURBHI SAROHA

2. SYLLABUS Set Operations Representations and Properties of Relations Equivalence Relations Partially Ordering

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 A c  (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. 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

10. 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

11. 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}

12. 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 .

13. 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

14. 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 .

15. Thank you 