Question

Determine the output of each of these circuits:

\({\bf{a)}}\)

\({\bf{b)}}\)

Short answer

\({\bf{(a)}}\)The output of the given circuit is \({\bf{(\bar x}} \oplus {\bf{y)}} \oplus {\bf{x}}\).

\({\bf{(b)}}\)The output of the given circuit is \({\bf{((x}} \oplus {\bf{y)}} \oplus {\bf{(\bar x}} \oplus {\bf{z))}} \oplus {\bf{(\bar y}} \oplus {\bf{\bar z)}}\).

Step-by-step solution

Step \({\bf{1}}\): Definition

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

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

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

The \({\bf{NOR}}\) operator \( \downarrow \) is \({\bf{1}}\) if both terms are \({\bf{0}}\).

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

The \({\bf{NAND}}\) operator \(\mid \) is \({\bf{1}}\) if either term is \({\bf{0}}\).

The Boolean operator \( \odot \) is \({\bf{1}}\) if both terms have the same value.

Finding the output

(a)

The output of an \({\bf{XOR}}\) gate is the \({\bf{XOR}}\) operator of the two inputs.

Therefore, the output is \({\bf{(\bar x}} \oplus {\bf{y)}} \oplus {\bf{x}}\).

 Step 3: Finding the output

(b)

The output of an \({\bf{XOR}}\) gate is the \({\bf{XOR}}\) operator of the two inputs.

Therefore, the output is \({\bf{((x}} \oplus {\bf{y)}} \oplus {\bf{(\bar x}} \oplus {\bf{z))}} \oplus {\bf{(\bar y}} \oplus {\bf{\bar z)}}\).