Chapter 4, Problem 43E

Explanation of Solution

Three-input AND gate:

• AND gate:
  • The AND gate performs the multiplication of its inputs.
  • It can accept more than two inputs and performs the product of its inputs to generate the output value.
  • The output of AND gate is nothing but the product of its input values.
• Boolean expression:
• Boolean expression is a form of mathematical notation to express the processing of gates.
• The Boolean expression that is used to represent the processing of three input AND gate is:

$\text{X}=\text{A }·\text{ B }·\text{ C}$

Where, A, B, and C are the inputs and X is the output.

• Truth table:
• Truth table is a table that contains all possible inputs and the equivalent output values to express the processing of gates.
• Truth table for three-input AND gate is shown below:

A	B	C	$\text{X}=\text{A }·\text{ B }·\text{ C}$
0	0	0	0
0	0	1	0
0	1	0	0
0	1	1	0
1	0	0	0
1	0	1	0
1	1	0	0
1	1	1	1

• In the above truth table:
  • When all the inputs are 0, then it generates the output as 0.

  $\text{X}=\text{0 }·\text{ 0 }·\text{ 0}=\text{0 }·\text{ 0}=\text{0}$

• When the inputs are A as 0, B as 0, and C as 1 then it generates the output as 0.

  $\text{X = 0 }·\text{ 0 }·\text{ 1 = 0 }·\text{ 1 = 0}$

• When the inputs are A as 0, B as 1, and C as 0 then it generates the output as 0