Question

A threshold gate represents a Boolean function. Find a Boolean expression for the Boolean function represented by this threshold gate.

Short answer

The Boolean function for the given gate is \({{\bf{x}}_{\bf{3}}}{\bf{ + }}\overline {{{\bf{x}}_{\bf{1}}}} {{\bf{x}}_{\bf{2}}}\).

Step-by-step solution

Definition

Let \({\bf{B = }}\left\{ {{\bf{0,1}}} \right\}\). Then \({{\bf{B}}^{\bf{n}}}{\bf{ = }}\left\{ {\left( {{{\bf{x}}_{\bf{1}}}{\bf{,}}{{\bf{x}}_{\bf{2}}}{\bf{, \ldots ,}}{{\bf{x}}_{\bf{n}}}} \right)\mid {{\bf{x}}_{\bf{i}}} \in {\bf{B}}} \right.\)for \(\left. {1 \le {\bf{i}} \le {\bf{n}}} \right\}\) is the set of all possible \({\bf{n}}\)-tuples of \({\bf{0's}}\) and \({\bf{1's}}\). The variable \({\bf{x}}\) is called a Boolean variable if it assumes values only from \({\bf{B}}\), that is, if its only possible values are \(0\) and \(1\). A function from \({{\bf{B}}^{\bf{n}}}\) to \({\bf{B}}\) is Boolean function and it has degree \({\bf{n}}\).

Using the figure to frame inequality

It needs to figure out which combinations of values for \({{\bf{x}}_{\bf{1}}}{\bf{,}}{{\bf{x}}_{\bf{2}}}\), and \({{\bf{x}}_{\bf{3}}}\) cause the inequality \({\bf{ - }}{{\bf{x}}_{\bf{1}}}{\bf{ + }}{{\bf{x}}_{\bf{2}}}{\bf{ + 2}}{{\bf{x}}_{\bf{3}}} \ge \frac{1}{2}\) to be satisfied. Clearly this will be true if \({{\bf{x}}_{\bf{3}}}{\bf{ = L}}\). If \({{\bf{x}}_{\bf{3}}}{\bf{ = 0}}\), then it will be true if and only if \({{\bf{x}}_{\bf{2}}}{\bf{ = 1}}\) and \({{\bf{x}}_{\bf{1}}}{\bf{ = 0}}\). Therefore, a Boolean expression for this function is \({{\bf{x}}_{\bf{3}}}{\bf{ + }}\overline {{{\bf{x}}_{\bf{1}}}} {{\bf{x}}_{\bf{2}}}\).