Question

Design a circuit that determines whether three or more of four individuals on a committee vote yes on an issue, where each individual uses a switch for the voting.

A threshold gate produces an output that is either or given a set of input values for the Boolean variables . A threshold gate has a threshold value , which is a real number, and weights , each of which is a real number. The output \({\bf{y}}\) of the threshold gate is \(1\) if and only if \({{\bf{w}}_{\bf{1}}}{{\bf{x}}_{\bf{1}}}{\bf{ + }}{{\bf{w}}_{\bf{2}}}{{\bf{x}}_{\bf{2}}}{\bf{ + L + }}{{\bf{w}}_{\bf{n}}}{{\bf{x}}_{\bf{n}}} \ge {\bf{T}}\). The threshold gate with threshold value \({\bf{T}}\) and weights \({{\bf{w}}_{\bf{1}}}{\bf{,}}{{\bf{w}}_{\bf{2}}}{\bf{, \ldots ,}}{{\bf{w}}_{\bf{n}}}\) is represented by the following diagram. Threshold gates are useful in modeling in neurophysiology and in artificial intelligence.

Short answer

The output of the gate is \({\bf{w x y + w x z + w y z + x y z}}\).

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 \( \bullet \) 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.

Using the Boolean sum and product

Let the votes of the four people be \({\bf{w x y}}\) and \({\bf{z}}\). This occurs if \({\bf{w x y,w x z,w y z}}\) or \({\bf{x y z}}\) is true (or if \({\bf{w x y z}}\) is true, but this will also be true if \({\bf{w x y,w x z,w y z}}\) and \({\bf{x y z}}\) are all true).

\({\bf{w x y + w x z + w y z + x y z}}\), Note: \({\bf{w x y}}\) means that \({\bf{w x}}\) and \({\bf{y}}\) were all "Yes" votes, while \({\bf{z}}\) could be a "Yes" or "No" vote. The output of an \({\bf{AND}}\) gate is the Boolean product of the two inputs. Therefore, the output of an \({\bf{OR}}\) gate is the Boolean sum of the two inputs, that is \({\bf{w x y + w x z + w y z + x y z}}\).