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Q9E

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Discrete Mathematics and its Applications

Found in: Page 818

Book edition 7th

Author(s) Kenneth H. Rosen

Pages 808 pages

ISBN 9780073383095

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Short Answer

What values of the Boolean variables \({\bf{x}}\) and \({\bf{y}}\) satisfy \({\bf{xy = x + y}}\)\(?\)

The value of x=0 and y=0 or x=1 and y=1 will solve the given equation \(xy = x + y\).

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Definition

The complement of an element: \({\bf{\bar 0 = 1}}\) and \({\bf{\bar 1 = }}0\)

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

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

Step 2: Using the Boolean product and sum

The given is \(xy = x + y\)

The function has three variables x, y and z. Each of these variables can take on the value of 0 or 1.

\(\begin{array}{*{20}{r}}x&y&{x \cdot y}&{x{\bf{ + }}y}\\0&0&0&0\\0&1&0&1\\1&0&0&1\\1&1&1&1\end{array}\)

One then notes that one obtains the same value in the last two columns of the table, if x=0 and y=1 or if x=1 and y=1.

Therefore, the value of x=0 and y=0 or x=1 and y=1 will solve the given equation \(xy = x + y\).

Most popular questions for Math Textbooks

In Exercises 1–5 find the output of the given circuit.

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Determine whether \({\bf{F}} \le {\bf{G}}\) or \({\bf{G}} \le {\bf{F}}\) for the following pairs of functions.

\(\begin{array}{c}{\bf{a) F(x,y) = x,G(x,y) = x + y}}\\{\bf{b) F(x,y) = x + y,G(x,y) = xy}}\\{\bf{c) F(x,y) = \bar x,G(x,y) = x + y}}\end{array}\)

Show that you obtain De Morgan's laws for propositions (in Table \(6\) in Section \(1.3\)) when you transform De Morgan's laws for Boolean algebra in Table \(6\) into logical equivalences.

Use \(K{\bf{ - }}\)maps to find simpler circuits with the same output as each of the circuits shown.

a)

b)

c)

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