• :00Days
• :00Hours
• :00Mins
• 00Seconds
A new era for learning is coming soon

Suggested languages for you:

Americas

Europe

Q7E

Expert-verified
Found in: Page 822

### Discrete Mathematics and its Applications

Book edition 7th
Author(s) Kenneth H. Rosen
Pages 808 pages
ISBN 9780073383095

# 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 ifa) $${\bf{x = }}\,{\bf{y = 1,}}\,{\bf{z = 0}}$$b) $${\bf{x = }}\,{\bf{y = }}\,{\bf{z = 0}}$$c) $${\bf{x = }}\,{\bf{z = 0,}}\,{\bf{y = 1}}$$

(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 $${\bf{x + }}\overline {\bf{y}} {\bf{ + z}}$$.

See the step by step solution

## Step 1: 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.

## Step 2: 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 $$\overline {\bf{x}} {\bf{ + }}\overline {\bf{y}} {\bf{ + z}}$$.

## Step 3: 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 $${\bf{x + y + z}}$$.

## Step 4: 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 $${\bf{x + }}\overline {\bf{y}} {\bf{ + z}}$$.