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Discrete Mathematics and its Applications
Found in: Page 844
Discrete Mathematics and its Applications

Discrete Mathematics and its Applications

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

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Short Answer

Construct a half adder using \(OR\) gates, \(AND\) gates, and inverters.

The sum is \({\bf{(x + y)(}}\overline {{\bf{xy}}} {\bf{)}}\) and carry\({\bf{xy}}\).

See the step by step solution

Step by Step Solution

Step 1: Definition

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

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

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

An inverter (Not gate) takes the complement of the input.

An \(AND\) gate takes the Boolean product of the input.

An \(OR\) gate takes the Boolean sum of the input.

Step 2: Circuit

The half adder determines the sum of two digits along with a carry. If \(x\) and \(y\) are the two digits, then their sum is \({\bf{(x + y)(}}\overline {{\bf{xy}}} {\bf{)}}\) and their carry is \({\bf{xy}}\) (which you cannot using an input and output table).

Therefore, the sum is \({\bf{(x + y)(}}\overline {{\bf{xy}}} {\bf{)}}\) and Carry\({\bf{xy}}\).

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