Question

Find a Boolean product of the Boolean variables x, y,and z, or their complements, that has the value 1 if and only if

a)x=y=0, z=1

b)x=0, y=1, z=0

c)x=0, y=z=1

d)x=y=z=0

Short answer

The Boolean products are

1. The product is\(\overline x \cdot \overline y \cdot z\).
2. The product is\(\overline x \cdot y \cdot \overline z \).
3. The product is\(\overline x yz\).
4. The product is \(\overline x \overline y \overline z \).

Step-by-step solution

Definition

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

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

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

(a) Find the result.

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

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

\(\begin{array}{l}\overline x = 1\\\overline y = 0\\z = 1\end{array}\)

The Boolean product of Boolean variable is 1 if all Boolean variable s 1.

\(\overline x \cdot \overline y \cdot z = 1\).

Thus, the Boolean product is \(\overline x \cdot \overline y \cdot z\).

(b) Determine the result. 

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

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

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

The Boolean product of Boolean variable is 1 if all Boolean variable s 1.

\(\overline x \cdot y \cdot \overline z = 1\).

Thus, the Boolean product is \(\overline x y\overline z \).

(c) Evaluate the result.

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

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

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

The Boolean product of Boolean variable is 1 if all Boolean variable s 1.

\(\overline x \cdot y \cdot z = 1\).

Thus, the Boolean product is \(\overline x yz\).

(d) Find the result. 

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

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

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

The Boolean product of Boolean variable is 1 if all Boolean variable s 1.

\(\overline x \cdot \overline y \cdot \overline z = 1\).

Therefore, the Boolean product is \(\overline x \overline y \overline z \).