Question

What values of the Boolean variables \({\bf{x}}\) and \({\bf{y}}\) satisfy \({\bf{xy = x + y}}\)\(?\)

Short answer

The value of x=0 and y=0 or x=1 and y=1 will solve the given equation \(xy = x + y\).

Step-by-step solution

Definition

The complement of an element: \({\bf{\bar 0 = 1}}\) and \({\bf{\bar 1 = }}0\)

The Boolean sum + or\(OR\)is 1 if either term is 1.

The Boolean product \( \cdot \) or \(AND\) is 1 if both terms are 1.

Using the Boolean product and sum

The given is \(xy = x + y\)

The function has three variables x, y and z. Each of these variables can take on the value of 0 or 1.

\(\begin{array}{*{20}{r}}x&y&{x \cdot y}&{x{\bf{ + }}y}\\0&0&0&0\\0&1&0&1\\1&0&0&1\\1&1&1&1\end{array}\)

One then notes that one obtains the same value in the last two columns of the table, if x=0 and y=1 or if x=1 and y=1.

Therefore, the value of x=0 and y=0 or x=1 and y=1 will solve the given equation \(xy = x + y\).