Let A = 1, 2, 3, , 10 . A relation R is defined on A by x R y if and only if (x, y) is a prime number. The number of elements in R is equal to :

Step-by-step solution

The prime numbers in the set A are p 2, 3, 5, 7 . For a given prime p , if (x, y) = p , then we can write x = pa and y = pb , where a, b 1, 2, , 10 p and (a, b) = 1 . Case 1: p = 2 Here, a, b 1, 2, 3, 4, 5 . The number of ordered pairs (a, b) with (a, b) = 1 is 2 _ k=1 ⁵ (k) - 1 . _ k=1 ⁵ (k) = (1) + (2) + (3) + (4) + (5) = 1 + 1 + 2 + 2 + 4 = 10 . Number of pairs for p=2 is 2(10) - 1 = 19 . Case 2: p = 3 Here, a, b 1, 2, 3 . _ k=1 ³ (k) = 1 + 1 + 2 = 4 . Number of pairs for p=3 is 2(4) - 1 = 7 . Case 3: p = 5 Here