Question

Deal with the Boolean algebra \(\{ 0,1\} \) with addition, multiplication, and complement defined at the beginning of this section. In each case, use a table as in Example \(8\).

Verify the zero property.

The Boolean operator \( \oplus \), called the \(XOR\) operator, is defined by \(1 \oplus 1{\bf{ = }}0,1 \oplus 0{\bf{ = }}1,0 \oplus 1{\bf{ = }}1\), and \(0 \oplus 0{\bf{ = }}0\).

Short answer

The given zero property \({\bf{x\bar x = 0}}\) is verified.

Step-by-step solution

Definition

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

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

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

Zero property, \({\bf{x\bar x = 0}}\).

Using the zero property

\({\bf{x}}\) can take on the value of \(0\) or \({\bf{1}}\).

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

The last two columns of the table are identical.

Therefore, itgives\({\bf{x\bar x = 0}}\).