Question

Find a Boolean product of Boolean sums of literals that has the value 0 if and only if \({\bf{x = y = 1}}\) and \({\bf{z = 0,x = z = 0}}\) and \({\bf{y = 1}}\), or \({\bf{x = y = z = 0}}\). (Hint: Take the

Boolean product of the Boolean sums found in parts (a), (b), and (c) in Exercise 7.)

Short answer

The required sum is\(\overline {{\bf{(x}}} {\bf{ + }}\overline {\bf{y}} {\bf{ + z)}}{\bf{.(x + y + z)}}{\bf{.(x + }}\overline {\bf{y}} {\bf{ + z)}}\) and product is 0.

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 term are 1.

check the result.

From the exercise 7 the result is from the part (a), (b), (c) receptively.

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

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

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

The Boolean product of Boolean sums of literals that has the value 0 if and only if \({\bf{x = y = 1}}\) and \({\bf{z = 0,x = z = 0}}\)and\({\bf{y = 1}}\), or \({\bf{x = y = z = 0}}\).

Then the Boolean product of the Boolean sums is \(\overline {{\bf{(x}}} {\bf{ + }}\overline {\bf{y}} {\bf{ + z)}}{\bf{.(x + y + z)}}{\bf{.(x + }}\overline {\bf{y}} {\bf{ + z)}}\).

Therefore, a Boolean product of Boolean sums of literals that has the value 0.