Consider the set {a, b, c} with binary operators + and * defined as follows. For +: a+a=b, a+b=a, a+c=c, b+a=a, b+b=b, b+c=c, c+a=a, c+b=c, c+c=b. For *: a*a=a, a*b=b, a*c=c, b*a=b, b*b=c, b*c=a, c*a=c, 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 and (b*x)+(c*y)=c. The number of solution(s) (i.e., pairs (x,y) that satisfy the equations) is A. 0 B. 1 C. 2 D. 3

GATE 2003 · Discrete Mathematics · Binary Operation · medium

Answer: 2 solution pairs: (b,c) and (c,b) (Option C)

1. Read the inner products from the * table: for each candidate (x,y) look up a*x, a*y for equation 1 and b*x, c*y for equation 2 directly from the * table
2. Combine with + and test against c for all 9 pairs: (b,c): Eq1 = (a*b)+(a*c) = b+c = c, Eq2 = (b*b)+(c*c) = c+b = c -> both c. (c,b): Eq1 = (a*c)+(a*b) = c+b = c, Eq2 = (b*c)+(c*b) = a+c = c -> both c. Every other pair fails at least one equation