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Let A = 1 , 2 , 3 , . ...100 . Let R be a relation on A defined by x , y ∈ R if and only if 2 x = 3 y . Let R 1 be a symmetric relation on A such that R ⊂ R 1 and the number of elements in R 1 is n . Then the minimum value of n is _______.

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Step-by-step solution

Given, 2 x = 3 y ⇒ x 3 = y 2 Now, let x 3 = y 2 = k ⇒ x = 3 k & y = 2 k Now, given set A = 1 , 2 , 3 , . . . . . . . , 100 So, for k ≤ 33 we get, ⇒ R = 3 , 2 , 6 , 4 , 9 , 6 , 12 , 8 , . ... 99 , 66 ⇒ n R = 33 Now, R 1 is symmetric relation on A and R ⊂ R 1 so, we need elements 3 , 2 , 6 , 4 , 9 , 6 , 12 , 8 , . ... 99 , 66 , 2 , 3 , 4 , 6 , 6 , 9 , 8 , 12 , . ... 66 , 99 in R 1 Hence, the minimum number of elements in R 1 is 33 + 33 = 66

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