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

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

*	a	b	c
a	a	b	c
b	b	c	a
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

We can write from given expression 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

a x a = a,          a x b = b,          a x c = c b x a = b,          b x b = c,          b x c = a c x a = c,          c x b = c,          c x c = b a x x + a x y = c                        --- (1) b x x + c x y = c             --- (2) First we see, these are the equation which give result c. a + c = c b + c = c c + b = c For equation (A), we find a x x = b  x = b We find a x x = a then x = a, a x y = c, y = c, x = a,b,c  a x y = c then y = c, a x x = c, x = c, x = c, y = a,b a x y = b, y = b for equation (B), we find b x x = b, c x y = c, b x x = a then x = c x = a, y = a, b    c x y = c then y = a, b b x x = c then    x = b    we find c x y = b then    y = c    x = a,b,c y = a,b,c so, total number of solution pairs (x,y) is 3.