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GATE | GATE-CS-2003 | Question 38

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Consider the set {a, b, c} with binary operators + and × defined as follows :

+ a b c × a b c
a b a c a a b c
b a b c b b c a
c a c b c c c b

For example, a + c = c, c + a = a, c × b = c and b × c = a. Given the following set of equations :

(a × x) + (a × y) = c
(b × x) + (c × y) = c

The number of solution(s) (i.e., pair(s) (x, y)) that satisfy the equations is :
(A) 0
(B) 1
(C) 2
(D) 3


Answer: (C)

Explanation: For this question, we will have to consider each case separately and check whether it satisfies both the conditions or not.

 

(a,a) ⇒ (a*a) + (a*a) = a + a = b ≠ c ⇒ (a,a) is not a solution.

(a,b) ⇒ (a*a) + (a*b) = a + b = a ≠ c ⇒ (a,b) is not a solution.

(a,c) ⇒ (a*a) + (a*c) = a + c = c

(b*a)  + (c*c) = b + b = b ≠ c ⇒ (a,c) is not a solution.

(b,a) ⇒ (a*b) + (a*a) = b + a = a ≠ c ⇒ (b,a) is not a solution.

(b,b) ⇒ (a*b) + (a*b) = b + b = b ≠ c ⇒ (b,b) is not a solution.

(b,c) ⇒ (a*b) + (a*c) = b + c = c

(b*b) + (c*c) = c + b = c ⇒ (b,c) is a solution

(c,a) ⇒ (a*c) + (a*a) = c + a = a ≠ c ⇒ (c,a) is not a solution.

(c,b) ⇒ (a*c) + (a*b) = c + b = c

(b*c) + (c*b) = a + c = c ⇒ (c,b) is a solution

(c,c) ⇒ (a*c) + (a*c) = c + c = b ≠ c ⇒ (c,c) is not a solution.

 

Thus, we have 2 solutions, (b,c) and (c,b).

Hence, C is the correct choice.

 

Please comment below if you find anything wrong in the above post.

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Last Updated : 28 Jun, 2021
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