Question

Find a Boolean sum containing either x or \(\overline {\bf{x}} \), either y or \(\overline {\bf{y}} \), and either z or \(\overline {\bf{z}} \) that has the value 0 if and only if

a) \({\bf{x = }}\,{\bf{y = 1,}}\,{\bf{z = 0}}\)

b) \({\bf{x = }}\,{\bf{y = }}\,{\bf{z = 0}}\)

c) \({\bf{x = }}\,{\bf{z = 0,}}\,{\bf{y = 1}}\)

Short answer

(a) The sum is\(\overline {\bf{x}} {\bf{ + }}\overline {\bf{y}} {\bf{ + z}}\).

(b)The sum is\({\bf{x + y + z}}\).

(c) The sum is\({\bf{x + }}\overline {\bf{y}} {\bf{ + z}}\).

Step-by-step solution

Definition:

The complements of an elements \(\overline {\bf{0}} {\bf{ = 1}}\) and \(\overline {\bf{1}} {\bf{ = 0}}\).

The Boolean sum + or OR is 1 if either term is 1.

The Boolean product (.) or AND is 1 if both terms are 1.

Find the result for \({\bf{x = }}\,{\bf{y = 1,}}\,{\bf{z = 0}}\).(a)

Here \({\bf{x = 1,}}\,{\bf{y = 1,}}\,{\bf{z = 0}}\).

If a Boolean variable is 0, the complement of the Boolean variable is 1.

\(\begin{array}{l}\overline {\bf{x}} {\bf{ = 0}}\\\overline {\bf{y}} {\bf{ = 0}}\\{\bf{z = 0}}\end{array}\)

The Boolean sum of the Boolean variable is 0 if all Boolean variable is 0.

\(\overline {\bf{x}} {\bf{ + }}\overline {\bf{y}} {\bf{ + z = 0}}\)

Thus, the Boolean product is \(\overline {\bf{x}} {\bf{ + }}\overline {\bf{y}} {\bf{ + z}}\).

Determine the result of \({\bf{x = }}\,{\bf{y = }}\,{\bf{z = 0}}\).(b)

Here, \({\bf{x = 0,}}\,{\bf{y = 1,}}\,{\bf{z = 0}}\).

If a Boolean variable is 0, then the complement of the Boolean variable is 1.

\(\begin{array}{l}{\bf{x = 0}}\\\overline {\bf{y}} {\bf{ = 0}}\\{\bf{z = 0}}\end{array}\)

The Boolean product of a Boolean variable is 0 if all Boolean variable is 0.

\({\bf{x + y + z = 0}}\).

Thus, the Boolean product is \({\bf{x + y + z}}\).

Evaluate the result for \({\bf{x = z = 0,}}\,{\bf{y = 1}}\).(c)

Here, \({\bf{x = 0,}}\,{\bf{y = 1,}}\,{\bf{z = 1}}\).

If a Boolean variable is 0, then the complement of the Boolean variable is 1.

\(\begin{array}{l}\overline {\bf{x}} {\bf{ = 1}}\\{\bf{y = 1}}\\{\bf{z = 1}}\end{array}\)

The Boolean product of Boolean variable is 0 if all Boolean variable is 0.

\({\bf{x + }}\overline {\bf{y}} {\bf{ + z = 0}}\)

Thus, the Boolean sum is \({\bf{x + }}\overline {\bf{y}} {\bf{ + z}}\).