Question

Show that \({\bf{x}} \oplus {\bf{y = y}} \oplus {\bf{x}}\).

Short answer

The given \(x \oplus y = y \oplus x\)is proved.

Step-by-step solution

Definition

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

The Boolean sum \( + \) or \(O{\bf{ }}R\) is \(1\) if either term is \(1\) .

The Boolean product or \(A{\bf{ }}N{\bf{ }}D\) is \(1\) if both terms are \(1\) .

The \(XOR\) operator \( \oplus \) is \(1\) if one of the terms is \(1\) (but not both).

Using the operator

\(x\)and \(y\) can both take on the value of \(0\) or \(1\) .

The \(XOR\) operator is \(1\) if one of the two elements (but not both) are \(1\) .

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

The last two columns of the table are identical.

Therefore, it get\(x \oplus y = y \oplus x\) .