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

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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 R 2.4 Application: Digital Logic Circuits2 INPUT P OUTPUT R 1 0

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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. 2.4 Application: Digital Logic Circuits3 AND P Q R INPU T P INPU T Q OUTPU T R 11 10 01 00

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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. 2.4 Application: Digital Logic Circuits4 OR Q P R INPU T P INPU T Q OUTPU T R 11 10 01 00

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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 Circuits5

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Give the output signals for the circuits if the input signals are as indicated. 2.4 Application: Digital Logic Circuits6

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Find the Boolean expression that corresponds to the circuit. 2.4 Application: Digital Logic Circuits7

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Construct circuits for the Boolean expression. P ( P Q) 2.4 Application: Digital Logic Circuits8

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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. 2.4 Application: Digital Logic Circuits9 PQRS 1110 1101 1010 1000 0111 0101 0010 0000

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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. 2.4 Application: Digital Logic Circuits10

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Logic Gates & Circuits. AND Gate Input AInput BOutput X 000 010 100 111 AND Logic Gate AND Truth Table X = A. B AND Boolean Expression.

Logic Gates & Circuits. AND Gate Input AInput BOutput X 000 010 100 111 AND Logic Gate AND Truth Table X = A. B AND Boolean Expression.

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