GATE CS 2003 Q38 — Sets, Relations and Functions

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

$$\begin{array}{|c|c|c|c|} \hline \textbf{+} & \textbf{a}& \textbf{b} &\textbf{c}\\ \hline \textbf{a} & \text{b}& \text{a} & \text{c} \\ \hline \textbf{b} & \text{a}& \text{b} & \text{c}\\ \hline \textbf{c} & \text{a}& \text{c} & \text{b}\\ \hline \end{array}\qquad \begin{array}{|c|c|c|c|} \hline \textbf{*} & \textbf{a}& \textbf{b} &\textbf{c}\\ \hline \textbf{a} & \text{a}& \text{b} & \text{c} \\ \hline \textbf{b} & \text{b}& \text{c} & \text{a}\\ \hline \textbf{c} & \text{c}& \text{c} & \text{b}\\ \hline \end{array}$$

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