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, zT.

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, zT.

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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