Notes On Types of Relations - CBSE Class 12 Maths
Relation A relation, R, from a non-empty set A to another non-empty set B, is a subset of A × B. The subset, R, is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B. Ex: A = {2, 4, 6, 9} and B = {4, 6, 18, 27, 54}. Find a relation R from A to B, such that for any a ∈ A and b ∈ B, 'a' is factor of 'b' and a < b. Solution: Since A = {2, 4, 6, 9} and B = {4, 6, 18, 27, 54}, we have to find the set of ordered pairs (a, b) such that a is factor of b and a < b. Let 2 ∈ A, 4 ∈ B, as 2 is a factor of 4 and 2 < 4. So (2, 4) is one such ordered pair. Likewise, (2, 6), (2, 18), (2, 54)… are other such ordered pairs. Thus, the required relation is R = {(2, 4), (2, 6), (2, 18), (2, 54), (6, 18), (6, 54,), (9, 18), (9, 27), (9, 54)}. Domain of R = {2, 6, 9} Range of R = {4, 6, 18, 27, 54} If (a, b) ∈ R, we say that a is related to b, and we denote it as a R b. If n (A) = p, n (B) = q, then n (A × B) = pq, and the total number of possible relations from set A to set B is 2pq . Empty Relation A relation R in a set A is called an empty relation, if no element of A is related to any element of A, i.e. R = ∅ ∈ A × A. Let R be a relation defined as R = {(x, y)/x+y ≤ 8} in A Choose any arbitrary ordered pair from A cross A. Here, all the ordered pairs satisfy the condition that the sum of their elements is less than or equal to 8. Hence, R = A × A. Universal Relation A relation R in a set A is called a universal relation or total relation, if every element of A is related to every element of A, i.e. R = A × A. An empty relation and the universal relation are called trivial relations. Different types of relations are: i) Reflexive ii) Symmetric iii) Transitive iv) Equivalence Reflexive Relation Let T be the set of all triangles in a plane. R is a relation in T defined by R = {(T1, T2): T1 is similar to T2} If (T1, T2) ∈ R, ⇒ T1 is similar to T1, which is true. ⇒ Every element of set R is related to itself. R in T is reflexive. A relation, R in T, is said to be reflexive if every element of set R is related to itself, i.e. (x, x) ∈ R, for every x ∈ T. Symmetric Relation Let T be the set of all triangles in a plane. R is a relation in T defined by R = {(T1, T2): T1 is similar to T2} Let (T1, T2) ∈ R, ⇒ T1 is similar to T2 ⇒ T2 is similar to T1 ⇒ (T1, T2) ∈ R A relation R in a set T is said to be symmetric if (x, y) ∈ R implies that (y, x) ∈ R  ∀ x, y ∈ T. Example: Relation R = {(1, 2), (2, 1), (1, 3)} in the set, {1, 2, 3}, is not symmetric.  '.'  (1, 3) ∈ R but (3, 1) ∉ R Note: It is wrong to say that it is "sometimes symmetric" or "is symmetric as far as 1 and 2 are concerned", since being symmetric is a property of the whole relation. Transitive Relation A Relation, R, in T is said to be transitive, if (x, y) ∈ R and (y, z) ∈ R imply that (x, z) ∈ R ∀ x, y, z ∈ T. Let T be the set of all triangles in a plane. R is a relation in T defined by R = {(T1, T2): T1 is similar to T2} Let T1, T2 and T3 ∈ T such that (T1 ,T2) ∈ R and (T2 ,T3) ∈ R. ⇒ T1 is similar to T2 and T2 is similar to T3 ⇒ T1 is similar to T3 ⇒ (T1 ,T3) ∈ R Hence, (T1 ,T2) ∈ R and (T2 ,T3) ∈ R ⇒ (T1 ,T3) ∈ R ∴ Relation, R, in T is transitive. Let T be the set of all triangles in a plane. R is a relation in T defined by R = {(T1, T2): T1 is similar to T2} The relation, R in T, is reflexive, symmetric and transitive. Relation R in T is an equivalence relation. Equivalence relation: A relation is said to be equivalence relation if it is reflexive, symmetric and transitive.

#### Summary

Relation A relation, R, from a non-empty set A to another non-empty set B, is a subset of A × B. The subset, R, is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B. Ex: A = {2, 4, 6, 9} and B = {4, 6, 18, 27, 54}. Find a relation R from A to B, such that for any a ∈ A and b ∈ B, 'a' is factor of 'b' and a < b. Solution: Since A = {2, 4, 6, 9} and B = {4, 6, 18, 27, 54}, we have to find the set of ordered pairs (a, b) such that a is factor of b and a < b. Let 2 ∈ A, 4 ∈ B, as 2 is a factor of 4 and 2 < 4. So (2, 4) is one such ordered pair. Likewise, (2, 6), (2, 18), (2, 54)… are other such ordered pairs. Thus, the required relation is R = {(2, 4), (2, 6), (2, 18), (2, 54), (6, 18), (6, 54,), (9, 18), (9, 27), (9, 54)}. Domain of R = {2, 6, 9} Range of R = {4, 6, 18, 27, 54} If (a, b) ∈ R, we say that a is related to b, and we denote it as a R b. If n (A) = p, n (B) = q, then n (A × B) = pq, and the total number of possible relations from set A to set B is 2pq . Empty Relation A relation R in a set A is called an empty relation, if no element of A is related to any element of A, i.e. R = ∅ ∈ A × A. Let R be a relation defined as R = {(x, y)/x+y ≤ 8} in A Choose any arbitrary ordered pair from A cross A. Here, all the ordered pairs satisfy the condition that the sum of their elements is less than or equal to 8. Hence, R = A × A. Universal Relation A relation R in a set A is called a universal relation or total relation, if every element of A is related to every element of A, i.e. R = A × A. An empty relation and the universal relation are called trivial relations. Different types of relations are: i) Reflexive ii) Symmetric iii) Transitive iv) Equivalence Reflexive Relation Let T be the set of all triangles in a plane. R is a relation in T defined by R = {(T1, T2): T1 is similar to T2} If (T1, T2) ∈ R, ⇒ T1 is similar to T1, which is true. ⇒ Every element of set R is related to itself. R in T is reflexive. A relation, R in T, is said to be reflexive if every element of set R is related to itself, i.e. (x, x) ∈ R, for every x ∈ T. Symmetric Relation Let T be the set of all triangles in a plane. R is a relation in T defined by R = {(T1, T2): T1 is similar to T2} Let (T1, T2) ∈ R, ⇒ T1 is similar to T2 ⇒ T2 is similar to T1 ⇒ (T1, T2) ∈ R A relation R in a set T is said to be symmetric if (x, y) ∈ R implies that (y, x) ∈ R  ∀ x, y ∈ T. Example: Relation R = {(1, 2), (2, 1), (1, 3)} in the set, {1, 2, 3}, is not symmetric.  '.'  (1, 3) ∈ R but (3, 1) ∉ R Note: It is wrong to say that it is "sometimes symmetric" or "is symmetric as far as 1 and 2 are concerned", since being symmetric is a property of the whole relation. Transitive Relation A Relation, R, in T is said to be transitive, if (x, y) ∈ R and (y, z) ∈ R imply that (x, z) ∈ R ∀ x, y, z ∈ T. Let T be the set of all triangles in a plane. R is a relation in T defined by R = {(T1, T2): T1 is similar to T2} Let T1, T2 and T3 ∈ T such that (T1 ,T2) ∈ R and (T2 ,T3) ∈ R. ⇒ T1 is similar to T2 and T2 is similar to T3 ⇒ T1 is similar to T3 ⇒ (T1 ,T3) ∈ R Hence, (T1 ,T2) ∈ R and (T2 ,T3) ∈ R ⇒ (T1 ,T3) ∈ R ∴ Relation, R, in T is transitive. Let T be the set of all triangles in a plane. R is a relation in T defined by R = {(T1, T2): T1 is similar to T2} The relation, R in T, is reflexive, symmetric and transitive. Relation R in T is an equivalence relation. Equivalence relation: A relation is said to be equivalence relation if it is reflexive, symmetric and transitive.

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