13.5.1 Combinational logic

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Logic gates are fundamental components in digital electronics. They are electronic switching circuits that output different signals depending on the combination of their inputs, which can either represent On ( $1$ , indicating current flow) or Off ( $0$ , no current flow). These gates are crucial in processing digital information and are widely used in devices like computer processors.

On ( $1$ ) – Indicates the presence of a current.
Off ( $0$ ) – Indicates the absence of a current.

Combinational Logic

When several logic gates are combined to produce a specific output based on the inputs, this setup is called combinational logic. The output depends solely on the current input values, without any memory of previous inputs. Combinational logic is essential for analysing and creating digital circuits with desired functionalities.

Truth Tables

To understand how a combination of inputs affects the output, truth tables are used. A truth table lists all possible input values and the corresponding output for each input combination. This is particularly helpful for simplifying and analysing digital circuits. For a circuit with $n$ inputs, there are $2^n$ possible input combinations. For example, with 4 inputs, there are $2^4 = 16$ combinations.

Types of Basic Logic Gates

AND Gate

Output is $1$ only if both inputs are $1$ ; otherwise, the output is $0$ .
Truth table:

A	B	Out
$0$	$0$	$0$
$0$	$1$	$0$
$1$	$0$	$0$
$1$	$1$	1

OR Gate

Output is $1$ if either input is $1$ ; otherwise, the output is $0$ .
Truth table:

A	B	Out
$0$	$0$	$0$
$0$	$1$	$1$
$1$	$0$	$1$
$1$	$1$	$1$

NOT Gate

Inverts the input; output is $1$ if input is $0$ , and vice versa.
Truth table:

A	Out
$0$	1
$1$	0

NAND Gate

Output is $0$ only if both inputs are $1$ ; otherwise, the output is $1$ .
This gate is essentially an AND gate followed by a NOT gate.
Truth table: | A | B | Out | |---|---|---| | $0$ | $0$ | $1$ | | $0$ | $1$ | $1$ | | $1$ | $0$ | $1$ | | $1$ | $1$ | $0$ |

NOR Gate

Output is $1$ only if both inputs are $0$ ; otherwise, the output is $0$ .
This gate is essentially an OR gate followed by a NOT gate.
Truth table: | A | B | Out | |---|---|---| | $0$ | $0$ | $1$ | | $0$ | 1 | $0$ | | $1$ | 0 | $0$ | | $1$ | 1 | $0$ |

EOR (XOR) Gate

Exclusive OR gate gives an output of $1$ if one input is $1$ and the other is $0$ .
Often used as a comparator.
Truth table: | A | B | Out | |---|---|---| | $0$ | 0 | $0$ | | $0$ | 1 | $1$ | | $1$ | 0 | $1$ | | $1$ | 1 | $0$ |

Boolean Algebra and Logic Circuit Simplification

Boolean algebra is a mathematical tool used to describe and optimise logic circuits. Using Boolean algebra, complex circuits can be represented with expressions, and unnecessary gates can be eliminated.

Key identities in Boolean algebra:

AND: $A \cdot B$
OR: $A + B$
NOT: $\overline{A}$

Key Boolean Laws

Associative Law

( $A \cdot B) \cdot C = A \cdot (B \cdot C)$
$(A + B) + C = A + (B + C)$

Commutative Law

$A \cdot B = B \cdot A$
$A + B = B + A$

Distributive Law

$A \cdot (B + C)$ = $(A \cdot B)$ + ( $A \cdot C)$
$A + (B \cdot C) = (A + B) \cdot (A + C)$

Simplifications

$A \cdot A = A$
$A + \overline{A} = 1$
$A \cdot 0 = 0$
$A + 0 = A$ These simplifications allow circuits to be designed with fewer gates, saving space and power.

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Example: Drawing Logic Circuits

Given a truth table or Boolean expression, you can construct a logic circuit. For example, to simplify $Z = (A \cdot \overline{B})$ + $($$\overline{A} \cdot B)$ , use: