Q. 44.0( 4 Votes )

# Find the inverse relation R^{-1} in each of the following cases:

i. R= {(1, 2), (1, 3), (2, 3), (3, 2), (5, 6)}

ii. R= {(x, y) : x, y N; x + 2y = 8}

iii. R is a relation from {11, 12, 13} to (8, 10, 12} defined by y = x – 3

Answer :

An inverse relation is the set of ordered pairs obtained by interchanging the first and second elements of each pair in the original relation. If the graph of a function contains a point (a, b), then the graph of the inverse relation of this function contains the point (b, a).

i. Given, R= {(1, 2), (1, 3), (2, 3), (3, 2), (5, 6)}

∴ R^{‑1} = {(2, 1), (3, 1), (3, 2), (2, 3), (6, 5)}

⇒ R^{‑1} = {(2, 1), (2, 3), (3, 1), (3, 2), (6, 5)}

ii. Given, R= {(x, y) : x, y N; x + 2y = 8}

Here, x + 2y = 8

⇒ x = 8 – 2y

As y N, Put the values of y = 1, 2, 3,…… till x N

On putting y=1, x = 8 – 2(1) = 8 – 2 = 6

On putting y=2, x = 8 – 2(2) = 8 – 4 = 4

On putting y=3, x = 8 – 2(3) = 8 – 6 = 2

On putting y=4, x = 8 – 2(4) = 8 – 8 = 0

Now, y cannot hold value 4 because x = 0 for y = 4 which is not a natural number.

∴ R = {(2, 3), (4, 2), (6, 1)}

R^{‑1} = {(3, 2), (2, 4), (1, 6)}

⇒ R^{‑1} = {(1, 6), (2, 4), (3, 2)}

iii. Given, R is a relation from {11, 12, 13} to (8, 10, 12} defined by y = x – 3

Here,

x {11, 12, 13} and y (8, 10, 12}

y = x – 3

On putting x = 11, y = 11 – 3 = 8 (8, 10, 12}

On putting x = 12, y = 12 – 3 = 9 ∉ (8, 10, 12}

On putting x = 13, y = 13 – 3 = 10 (8, 10, 12}

∴ R = {(11, 8), (13, 10)}

R^{‑1} = {(8, 11), (10, 13)}

Rate this question :

Write the relation R = {(x, x^{3}): x is a prime number less than 10} in roster form.

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 is relatively prime to y. Express R as a set of ordered pairs and determine its domain and range.

RD Sharma - MathematicsWrite the following relations as the sets of ordered pairs:

A relation R from the set {2, 3, 4, 5, 6} to the set {1, 2, 3} defined by x = 2y.

RD Sharma - MathematicsDetermine the domain and range of the relation R defined by

R= {(x, x^{3}): x is a prime number less than 10}

Let A = {3, 5} and B = {7, 11}. Let R = {(a, b): a A, b B, a-b is odd}. Show that R is an empty relation from into B.

RD Sharma - MathematicsLet A = {1, 2} and B={3, 4}. Find the total number of relations from A into B.

RD Sharma - Mathematics