Question

Suppose that there are five members on a committee, but that Smith and Jones always vote the opposite of Marcus. Design a circuit that implements majority voting of the committee using this relationship between votes.

Short answer

The result of the majority voting is \({\bf{xyz + (x + y)\bar z}}\).

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 \({\bf{1}}\).

Assuming the minterm

Five people among which are Smith, Jones and Marcus. Smith and Jones always vote opposite to Marcus. Let \({\bf{x}}\) and \({\bf{y}}\) be the votes of the two unnamed people and let \({\bf{z}}\) represents the vote of Marcus.

If \({\bf{x}}\) and \({\bf{y}}\) are the same vote as \({\bf{z}}\), then \(x,y,z\) will form the majority (as Smith and Jones vote the opposite) and this then corresponds with the minterm\({\bf{xyz}}\). If at least one of the unnamed people vote differently from \({\bf{z}}\), then this person will form the majority with Smith and Jones. \({\bf{z}}\) thus, has to be opposite to \({\bf{x}}\) or \({\bf{y}}\), which can be represented \({\bf{(x + y)\bar z}}x\) or \({\bf{y}}\), and not \({\bf{z}}\). Thus, the result of the majority voting is then \({\bf{xyz + (x + y)\bar z}}\).

Using the Boolean sum

The complement is represented by an inverter in a circuit. The Boolean sum is represented by an \(OR\) gate in a circuit. The Boolean product is represented by an \(AND\) gate in a circuit.