## About "Finding Inverse of a Relation From the Given Relation"

Finding Inverse of a Relation From the Given Relation :

Here we are going to see some example problems to understand finding inverse of a relation.

## Finding Inverse of a Relation From the Given Relation - Examples

Let A and B be two sets and R be a relation of a set to a set B. Then the inverse of R denoted by R-1 is a relation from B to A and is defined by R-1  =  {(b, a) : (a, b) ∈ R}

Clearly (a, b) ∈ R <=> R-1

Also, (i)  Domain of R = Range(R-1) and

(ii)  range (R) = Domain(R-1)

## Finding Inverse of a Relation From the Given Relation - Examples

Question 1 :

Let A be the set of first ten natural numbers and let R be a relation defined by (x, y) ∈ R <=> x + 2y  =  10 where R = {(x,y), x ∈ A, y ∈ A and x + 2y  =  10. Express R and R-1 as the set of ordered pairs. Also determine

(i)  Domain of R and R-1

(ii)  Range of R and R-1

Solution :

To find the set of ordered pairs in the relation, let us apply the values of x one by one in the given function.

x + 2y  =  10

y  =  (10 - x)/2

A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

 If x = 1y = 9/2 ∉ A If x = 2y = 8/2 y = 4 If x = 3y = 7/2 ∉ A If x = 4y = 6/2 y = 3 If x = 6y = 4/2 y = 2 If x = 8y = 2/2 y = 1 If x = 10y = 0/2 y = 0

R = {(2, 4) (4, 3) (6, 2) (8, 1) (10, 0)}

Domain of R  =  {2, 4, 6, 8, 10}

Range of R  =  {4, 3, 2, 1, 0}

R-1  =  {(4, 2) (3, 4) (2, 6) (1, 8) (0, 10)}

Domain of R-1  =  {4, 3, 2, 1, 0}

Range of R-1  =  {2, 4, 6, 8, 10}

Question 2 :

A relation R is defined from a set A  =  {2, 3, 4, 5} to a set B  = {3, 6, 7, 10} as follows (x, y)  R => x divides y. Express R as the set of ordered pairs and determine the domain and range of R also find R-1.

Solution :

R  =  {(2, 6) (2, 10) (3, 3) (3, 6) (5, 10)}

Domain of R  =  {2, 3, 5}

Range of R  =  {3, 6, 10}

R-1  =  {(6, 2) (10, 2) (3, 3) (6, 3) (10, 5)}

Question 3 :

Let R be relation in N defined by (x, y) ∈ R <=> x + 2y  =  8. Express R and R-1 as a set of ordered pairs.

Solution :

x = 0, 1, 2, ...........

y  =  (8 - x)/2

 If x = 1y = 7/2 ∉ N If x = 2y = 6/2 y = 3 ∈ N If x = 4y = 4/2y = 2 ∈ N If x = 6y = 2/2y = 1 ∈ N If x = 8y = 0/2 y  = 0 ∉ N

R = {(2, 3) (4, 2) (6, 1)}

R-1  =  {(3, 2) (2, 4) (1, 6)} After having gone through the stuff given above, we hope that the students would have understood "Finding Inverse of a Relation From the Given Relation".

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