Chapter 4, Problem 35E

Explanation of Solution

Gates:

• A gate is a device which is used to perform the basic logical operation.
• It accepts one or more input signals and generates single output signal.
• It contains several types of gates.
• Six types of gates are most commonly used. They are:
  • NOT gate
  • AND gate
  • OR gate
  • XOR gate
  • NAND gate
  • NOR gate
• NOT gate:
  • A NOT gate accepts only one input value to generate single output value.
  • When the input is 0, then the output is 1. The output of the NOT gate is the inversion of its input.
  • So, it is also referred to as inverter.
  • It is represented in three ways:
    • In “Boolean expression”, the NOT operation is expressed using the “$\text{'}$” mark or “$\overline{}$” horizontal bar symbol.

  $\text{X = A'}$     or      $\text{X = }\overline{\text{A}}$

Where A is the input and X is the output.

• The “logic diagram” for NOT gate takes A as input and generates a single X as output:

• The “Truth table” for the NOT gate:

A	$\text{X = }\overline{\text{A}}$
0	1
1	0

• AND gate:
  • An AND gate accepts two inputs and generates a single output.
  • When both the inputs are 1, then the output is 1. Otherwise, the output is 0.
  • It is represented in three ways:
    • In “Boolean expression”, the AND operation is expressed using the “.” dot operator or “*” asterisk operator.

  $\text{X = A }\cdot \text{ B}$     or      $\text{X = A * B}$

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

• The “logic diagram” for AND gate takes two inputs and generates a single output:

• The “Truth table” for the AND gate:

A	B	$\text{X = A }\cdot \text{ B}$
0	0	0
0	1	0
1	0	0
1	1	1

• OR gate:
  • An OR gate accepts two inputs and generates a single output.
  • When both the inputs are 0, then the output is 0. Otherwise, the output is 1.
  • It is represented in three ways:
    • In “Boolean expression”, the OR operation is expressed using the “+” plus sign.

      $\text{X}=\text{A}+\text{B}$

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

• The “logic diagram” for OR gate takes two inputs and generates a single output:

• The “Truth table” for the OR gate:

A	B	$\text{X}=\text{A}+\text{B}$
0	0	0
0	1	1
1	0	1
1	1	1

• XOR gate:
  • An XOR gate accepts two inputs and produces a single output.
  • When both the inputs are same, then the output is 0. Otherwise, the output is 1.
  • It is represented in three ways:
    • In “Boolean expression”, the XOR operation is expressed by using the “$\oplus$” symbol.

      $\text{X = A }\oplus \text{ B}$

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

• The “logic diagram” for XOR gate takes two inputs and generates a single output:

• The “Truth table” for the XOR gate:

A	B	X = A$\oplus$B
0	0	0
0	1	1
1	0	1
1	1	0

• NAND gate:
  • The NAND gate accepts two inputs and produces a single output.
  • It produces the opposite results of an AND gate.
  • If both the inputs are 1, then the output is 0. Otherwise, the output is 1.
  • It is represented in three ways:
    • The “Boolean expression” for NAND gate:
      • No specific symbol is used to express the NAND operation.
      • The expression for NAND is the negation of an AND operation.

$\text{X = (A }·\text{ B)'}$  or   $\text{X = }\overline{\text{(A }·\text{ B)}}$

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

• The “logic diagram” for NAND gate:
  • It takes two inputs and generates a single output.

• The “Truth table” for the NAND gate:

A	B	$\text{X =}\overline{\text{ (A }·\text{ B)}}$
0	0	1
0	1	1
1	0	1
1	1	0

• NOR gate:
  • The NOR gate accepts two inputs and produces a single output.
  • It produces the opposite results of an OR gate.
  • If both the inputs are 0, then only the output is 1. Otherwise, the output is 0.
  • It is represented in three ways:
    • In “Boolean expression”,
      • No specific symbol is used to express the NOR operation.
      • The expression for NOR is the negation of an OR operation.

        $\text{X = (A + B)'}$ or $\text{X = }\overline{\text{(A + B)}}$

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

• The “logic diagram” for NOR gate takes two inputs and generates a single output:

• The “Truth table” for the NOR gate:

A	B	$\text{X =}\overline{\text{ (A + B)}}$
0	0	1
0	1	0
1	0	0
1	1	0