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Logic Gates, Boolean Algebra and Truth Tables


Logic Gates, Boolean Algebra and Truth Tables

Boolean Algebra is the mathematical foundation of digital circuits. Boolean Algebra specifies the relationship between Boolean variables which is used to design combinational logic circuits using Logic Gates. The truth table shows a logic circuit's output response to all of the input combinations.
  • Boolean Algebra

    • A Boolean Variable takes the value of either 0 (False) or 1 (True).
    • Symbols are used to represent Boolean variables e.g. A, B, C, X, Y, Z
    • There are three basic logic operations AND, OR, NOT
    • The Boolean Operators are • + ‾
      • A + B means A OR B
      • A • B means A AND B
      • A means NOT A
    • Nodes in a circuit are represented by Boolean Variables
  • Logic Gates

    • Logic gates are electronic circuits that implement the basic functions of Boolean Algebra. There is a symbol for each gate
      Z = A + B
      ABZ
      000
      011
      101
      111
            
    • Logic levels (0 or 1) are represented by means of a voltage level.
      • High voltage (5V, 3.3V, 2.5 V, etc.) is 1
      • Low voltage (0V) is 0
  • Truth Table

    The Truth Table shows the values of the circuit output for all input values.

Equivalent Circuits

Fig 1 is an example of four circuits that are equivalent because their truth tables are identical.
Equivalent NAND circuits
  • (a) is a NOT AND circuit
  • (b) is a NAND gate which gets its name from its Not AND configuration. It uses a bubble to represent a NOT gate at its output
  • (c) implements the NAND function using OR and NOT gates
  • (d) is another way to draw (c) using a bubble to represent a NOT gate at its input

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