Question

Construct a half adder using \(OR\) gates, \(AND\) gates, and inverters.

Short answer

The sum is \({\bf{(x + y)(}}\overline {{\bf{xy}}} {\bf{)}}\) and carry\({\bf{xy}}\).

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\( \cdot \) or \(AND\) is \({\bf{1}}\) if both terms are \({\bf{1}}\).

An inverter (Not gate) takes the complement of the input.

An \(AND\) gate takes the Boolean product of the input.

An \(OR\) gate takes the Boolean sum of the input.

Circuit

The half adder determines the sum of two digits along with a carry. If \(x\) and \(y\) are the two digits, then their sum is \({\bf{(x + y)(}}\overline {{\bf{xy}}} {\bf{)}}\) and their carry is \({\bf{xy}}\) (which you cannot using an input and output table).

Therefore, the sum is \({\bf{(x + y)(}}\overline {{\bf{xy}}} {\bf{)}}\) and Carry\({\bf{xy}}\).