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Q7E

Expert-verifiedFound in: Page 822

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 if**

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

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

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

**(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 **\({\bf{x + }}\overline {\bf{y}} {\bf{ + z}}\).

** **

** **

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.

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

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

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

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