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Discrete Structures

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. NOT-Gate.

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Discrete Structures

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  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 R 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. P AND R Q 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. P OR R Q 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: • Never combine two input wires. • A single input wire can be split partway and used as input for two separate gates. • An output wire can be used as an input. • 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. 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. 2.4 Application: Digital Logic Circuits

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