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Solution


Q.

Let A = {–3, –2, –1, 0, 1, 2, 3} and R be a relation on A defined by xRy if and only if 2x – y ∈ {0, 1}. Let l be the number of elements in R. Let m and n be the minimum number of elements required to be added in R to make it reflexive and symmetric relations, respectively. Then l + m + n is equal to :          [2025]
1 17  
2 16  
3 18  
4 15  

Ans.

(1)

We have, A = {–3, –2, –1, 0, 1, 2, 3}, R is defined on A as xRy such that 2x – y ∈ {0, 1}.

i.e., 2x – y = 0 or 2x – y = 1

∴   R = {(0, 0), (–1, –2), (1, 2), (0, –1), (2,3), (1, 1), (–1, –3)} i.e., l = 7

For R to be reflexive, i.e., we need 5 more elements {(2, 2), (–1, –1), (3, 3), (–3, –3), (–2, –2)} so m = 5 and for R to be symmetric, we need 5 more elements {(–2, –1), (2, 1), (–1, 0), (3, 2), (–3, –1)}, so n = 5.

∴   l + m + n = 7 + 5 + 5 = 17.