Question

Exercises 14-23 deal with the Boolean algebra \(\left\{ {{\bf{0,1}}} \right\}\) with addition, multiplication, and complement defined at the beginning of this section. In each case, use a table as in Example \(8\).

16. Verify the identity laws.

Short answer

The identity law is verified by using the Boolean sum and product.

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 identity law

Identity laws

\(\begin{array}{c}x + 0 = x\\x \cdot 1 = x\end{array}\)

x can take on the value of 0 or 1

The Boolean sum is 1 if one of the two elements (or both) are 1.

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

One notes that the last two columns of the table are identical, which implies \(x + 0 = x\).

Using the identity law

The Boolean product is 1 if both elements are 1.

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

One notes that the last two columns of the table are identical, which implies \(x \cdot 1 = x\).

Hence, the identity law is verified by using the Boolean sum and product.