Digital Logic Systems

Combinational Circuits

Basic Gates&

Truth Tables

Basic Gates

AND Gate OR Gate NOT Gate

More Gates

NAND Gate NOR Gate BUF Gate

More Gates

XNOR GateXOR Gate

n-Input Gates

3-Input XOR Gate

5-Input NOR Gate 5-Input AND Gate

4-Input OR Gate

Definitions

AND It gives a logical output true only if all the inputs are true

OR It gives a logical output true if any of the inputs is true

XOR It gives a logical output true only if an odd-number of inputs is true

NOT It gives a logical output true if the input is false and vice versa

Truth Table

A truth table is a tabular procedure to express the relationship of the outputs to the inputs of a Logical System

Truth Tables for Gates

a b fAND

0 0 00 1 01 0 01 1 1

a b fOR

0 0 00 1 11 0 11 1 1

a fNOT

0 11 0

AND Operation OR Operation

NOT Operation

AND Gate OR Gate NOT Gate

Truth Tables for Gates

a b fNAND

0 0 10 1 11 0 11 1 0

a b fNOR

0 0 10 1 01 0 01 1 0

a fBUF

0 01 1

NAND Operation NOR Operation

BUF Operation

NAND Gate NOR Gate BUF Gate

Truth Tables for Gates

a b fXOR

0 0 00 1 11 0 11 1 0

a b fXNOR

0 0 10 1 01 0 01 1 1

XOR Operation XNOR Operation

XNOR GateXOR Gate

A Bubble Implies a Logical Inversion

Bubbles can be replaced by NOT Gates to get logically equivalent

circuits

Bubbles

Generate tables for all combinations of bubbles and a XOR gate

Gate Equivalence

===

Gate Equivalence

== ?

Gate Equivalence

= =

Switching Expressions

Basic Switching Expressions

AND f = a . b

OR f = a + b

NOT f = a’f = ā

Is there an expression for XOR operation?

Switching Expressions

Switching Expressions

Switching Expressions

f1 = a . b’f2 = (a + b)’

Switching Expressions

Switching Expressions

Switching Expressions

f = ?

Switching Expressions

f = m + n

n = a’ . bm = a . b’

Switching Expressions

f = (a . b’) + (a’ . b)This is the equivalent circuit and equivalent

expression for a XOR operation

From Digital Design, 5th Edition by M. Morris Mano and Michael Ciletti

Switching Expressions

Switching Expressions

Switching Expressions

f1 = a . bf2 = a ^ bf2 = (a . b’) + (a’ . b)

Switching Expressions

x y z p = x ^ y g = x . y m = p . z s = p ^ z c = m + g 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1

x y z p = x ^ y g = x . y m = p . z s = p ^ z c = m + g 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 1 1 1 0 1 0 0 1 0 1 0 1 1 0 1 1 0 0 1 1 1 1 0 1

x y z p = x ^ y g = x . y m = p . z s = p ^ z c = m + g 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1 1 1 0 1 1 0 0 1 0 0 1 0 1 1 0 1 1 1 0 0 1 0 1 1 1 0 1 0

x y z p = x ^ y g = x . y m = p . z s = p ^ z c = m + g 0 0 0 0 0 0 0 00 0 1 0 0 0 1 00 1 0 1 0 0 1 00 1 1 1 0 1 0 11 0 0 1 0 0 1 01 0 1 1 0 1 0 11 1 0 0 1 0 0 11 1 1 0 1 0 1 1

x y z s c0 0 0 0 00 0 1 1 00 1 0 1 00 1 1 0 11 0 0 1 01 0 1 0 11 1 0 0 11 1 1 1 1

s = sc = m + g

s = sc = m + g m = p . z

g = g

s = p ^ z

s = sc = m + g m = p . z

g = g

p = x ^ y g = x . y

s = p ^ z

s = sc = m + g

p = x ^ y g = x . y m = (x ^ y) . z

g = g

s = (x ^ y) ^ z

s = (x ^ y) ^ zc = ((x ^ y) . z) + (x . y)

p = x ^ y g = x . y m = (x ^ y) . z

g = g

s = (x ^ y) ^ z

s = (x ^ y) ^ zc = ((x ^ y) . z) + (x . y)

s = ((x . y’) + (x’ . y)) ^ zc = (((x . y’) + (x’ . y)) . z) + (x . y)

s = (((x . y’) + (x’ . y))’ . z) + (((x . y’) + (x’ . y)) . z’)c = (((x . y’) + (x’ . y)) . z) + (x . y)

Procedure

To obtain the output functions from a logic diagram, proceed as follows:

1. Label with arbitrary symbols all gate outputs that are a function of the input variables. Obtain the Boolean Functions for each gate.

2. Label with other arbitrary symbols those gates that are a function of input variables and/or preciously labeled gates. Find the Boolean functions of these gates.

3. Repeat the process in step 2 until all the outputs of the circuit are obtained.4. By repeated substitution of previously defined functions, obtain the output

Boolean functions in terms of input variables only.