15.2 Boolean Algebra and Logic Circuits

This section explores Boolean algebra, a fundamental concept in digital logic, and its application to designing logic circuits. We will cover basic Boolean operations, algebraic laws, and how these are translated into circuit diagrams.

1. Basic Boolean Operations

Boolean algebra deals with logical values: True (represented by 1) and False (represented by 0). The basic Boolean operations are:

• AND : The output is 1 only if both inputs are 1. Symbol: AND gate symbol. Table:

  Input A	Input B	Output
  0	0	0
  0	1	0
  1	0	0
  1	1	1

• OR : The output is 1 if either input is 1. Symbol: OR gate symbol. Table:

  Input A	Input B	Output
  0	0	0
  0	1	1
  1	0	1
  1	1	1

• NOT: The output is the logical inverse of the input. Symbol: NOT gate symbol. Table:

  Input	Output
  0	1
  1	0

2. Boolean Algebra Laws

Boolean algebra provides a set of laws that allow us to simplify Boolean expressions. These laws are crucial for designing efficient logic circuits.

• Commutative Law: A AND B = B AND A, A OR B = B OR A
• Associative Law: (A AND B) AND C = A AND (B AND C), (A OR B) OR C = A OR (B OR C)
• Distributive Law: A AND (B OR C) = (A AND B) OR (A AND C), A OR (B AND C) = (A OR B) AND (A OR C)
• Identity Law: A AND 1 = A, A OR 0 = A
• Complement Law: A AND NOT A = 0, A OR NOT A = 1
• Idempotent Law: A AND A = A, A OR A = A
• Absorption Law: A AND (A OR B) = A, A OR (A AND B) = A
• De Morgan's Laws: NOT(A AND B) = OR(NOT A OR NOT B), NOT(A OR B) = AND(NOT A AND NOT B)

3. Logic Gates and Circuit Diagrams

Logic gates are electronic circuits that implement Boolean functions. They are represented by standard symbols and can be combined to create more complex circuits.

Gate	Symbol	Boolean Function
AND	AND gate symbol	$A \cdot B $
OR	OR gate symbol	$A + B$
NOT	NOT gate symbol	$ \overline{A}$ or $A'$
NAND	NAND gate symbol	$ \overline{A+B} $
NOR	NOR gate symbol	$ \overline{A \cdot B} $
XOR	XOR gate symbol	$ A \oplus B $

For example, a NAND gate is created by combining an AND gate with a NOT gate. The output of the AND gate is then inverted by the NOT gate.

Complex logic circuits can be constructed by combining these basic gates. Circuit diagrams use standard symbols to represent these gates and their interconnections.

4. Truth Tables and Circuit Design

Truth tables are used to define the behavior of a Boolean function. The truth table lists all possible input combinations and their corresponding output values. These truth tables are then used to design logic circuits.

By analyzing the truth table, we can determine the Boolean expression that represents the function and then translate that expression into a circuit diagram using logic gates.