2.4 Application: Digital Logic Circuits 1

Discrete Structures

Chapter 2: The Logic of Compound Statements

2.4 Application: Digital Logic Circuits

Only connect! – E. M. Forster, 1879 – 1970

Howards End, 1910

2.4 Application: Digital Logic Circuits 2

NOT-Gate

A NOT-gate (or inverter) is a circuit with one input signal and one output signal. The NOT-gate signals correspond exactly to the logical connector ~ if the symbol 1 is identified with T and the symbol 0 is identified with F.

P NOT RINPUT

POUTPUT

R

1

0

2.4 Application: Digital Logic Circuits 3

AND-Gate

• An AND-gate is a circuit with two input signals and one output signal. The AND-gate signals correspond exactly to the logical connector if the symbol 1 is identified with T and the symbol 0 is identified with F.

ANDPQ

R

INPUT P

INPUTQ

OUTPUT R

1 1

1 0

0 1

0 0

2.4 Application: Digital Logic Circuits 4

OR-Gate

• The OR-gate also has two input signals and one output signal. The AND-gate signals correspond exactly to the logical connector if the symbol 1 is identified with T and the symbol 0 is identified with F.

ORQP

R

INPUT P

INPUTQ

OUTPUT R

1 1

1 0

0 1

0 0

2.4 Application: Digital Logic Circuits 5

Rules for Combinational Circuit

• Gates can be combined into circuits in a variety of ways. When we follow the rules below, we create a combinational circuit, one whose output at anytime is determined entirely by its input at that time without regard to previous inputs.

• Rules:1. Never combine two input wires.

2. A single input wire can be split partway and used as input for two separate gates.

3. An output wire can be used as an input.

4. No output of a gate can eventually feed back into that gate.

2.4 Application: Digital Logic Circuits 6

Example – pg. 76 #2

• Give the output signals for the circuits if the input signals are as indicated.

2.4 Application: Digital Logic Circuits 7

Example – pg. 76 #12

• Find the Boolean expression that corresponds to the circuit.

2.4 Application: Digital Logic Circuits 8

Example – pg. 76 # 15

• Construct circuits for the Boolean expression.

P (P Q)

2.4 Application: Digital Logic Circuits 9

Example – pg. 76 # 21

• For the given table, construct (a) a Boolean expression having the given table as its truth table and (b) a circuit having the given table as its input/output table.

P Q R S

1 1 1 0

1 1 0 1

1 0 1 0

1 0 0 0

0 1 1 1

0 1 0 1

0 0 1 0

0 0 0 0

2.4 Application: Digital Logic Circuits 10

Example – pg. 77 #27

• Use the properties listed in Theorem 2.1.1 to show that each pair of circuits have the same input/output table. Find the Boolean expressions for the circuits and show that they are logically equivalent when regarded as statement forms.