Chapter 2: The Logic of Compound Statements 2.4 Application: Digital Logic Circuits 1 Only connect!...
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Discrete Structures Chapter 2: The Logic of Compound Statements 2.4 Application: Digital Logic Circuits 1 2.4 Application: Digital Logic Circuits Only connect! – E. M. Forster, 1879 – 1970 Howards End, 1910
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Transcript of Chapter 2: The Logic of Compound Statements 2.4 Application: Digital Logic Circuits 1 Only connect!...
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.
