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Found in: Page 844

### Discrete Mathematics and its Applications

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

# 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 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$${\bf{xy}}$$.