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Solution


Q.

Let A={-2,-1,0,1,2,3,4}. Let R be a relation on A defined by xRy if and only if 2x+y≤2. 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:                 [2026]
1 33  
2 32  
3 35  
4 34  

Ans.

(1)

R = {(-2, a), (-1, b), (0, c), (1, d), (2, e)}

a = {-2, -1, 0, 1, 2, 3, 4};  
b = {-2, -1, 0, 1, 2, 3, 4}  
c = {-2, -1, 0, 1, 2};  
d = {-2, -1, 0};  
e = {-2}

∴ No. of elements in R  
= 7 + 7 + 5 + 3 + 1 = 23 = ℓ

Minimum number of element to be added to make it reflexive = m = 4  
⇒ {(1, 1), (2, 2), (3, 3), (4, 4)}

Minimum number of element to be added to make it symmetric = n = 6  

⇒ R = {(3, -2), (4, -2), (2, -1), (2, 0), (3, -1), (4, -1)}

∴ ℓ + m + n = 23 + 4 + 6 = 33