Question

Write the following relations as the sets of ordered pairs:
(i) A relation R from the set {2, 3, 4, 5, 6} to the set {1, 2, 3} defined by x = 2y.

(ii) A relation R on the set {1, 2, 3, 4, 5, 6, 7} defined by (x, y) ∈ R ⇔ x is relatively prime to y.

(iii) A relation R on the set {0, 1, 2,…,10} defined by 2x + 3y = 12.

(iv) A relation R form a set A = {5, 6, 7, 8} to the set B = {10, 12, 15, 16, 18} defined by (x, y) R x divides y.

Answer 1

(i) Suppose A = {2, 3, 4, 5, 6} and B = {1, 2, 3}

Given as, x = 2y where y = {1, 2, 3}

If, y = 1, x = 2(1) = 2

If, y = 2, x = 2(2) = 4

If, y = 3, x = 2(3) = 6

∴ R = {(2, 1), (4, 2), (6, 3)}

(ii) Given as

(x, y) R x is relatively prime to y

Since,

2 is the co-prime to 3, 5 and 7.

3 is the co-prime to 2, 4, 5 and 7.

4 is the co-prime to 3, 5 and 7.

5 is the co-prime to 2, 3, 4, 6 and 7.

6 is the co-prime to 5 and 7.

7 is the co-prime to 2, 3, 4, 5 and 6.

∴ R = {(2, 3), (2, 5), (2, 7), (3, 2), (3, 4), (3, 5), (3, 7), (4, 3), (4, 5), (4, 7), (5, 2), (5, 3), (5, 4), (5, 6), (5, 7), (6, 5), (6, 7), (7, 2), (7, 3), (7, 4), (7, 5), (7, 6), (7, 7)}

(iii) Given as

(x, y) R 2x + 3y = 12

Here x and y = {0, 1, 2,…, 10}

2x + 3y = 12

2x = 12 – 3y

x = (12-3y)/2

If, y = 0, x = (12-3(0))/2 = 12/2 = 6

If, y = 2, x = (12-3(2))/2 = (12-6)/2 = 6/2 = 3

If, y = 4, x = (12-3(4))/2 = (12-12)/2 = 0/2 = 0

∴ R = {(0, 4), (3, 2), (6, 0)}

(iv) Given as

(x, y) R x divides y

Here, x = {5, 6, 7, 8} and y = {10, 12, 15, 16, 18}

Since,

5 divides 10 and 15.

6 divides 12 and 18.

7 divides none of the value of set B.

8 divides 16.

Thus, R = {(5, 10), (5, 15), (6, 12), (6, 18), (8, 16)}