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• Boolean Algebra is a mathematical representation of logic gates.
• Any logic circuit can be condensed into Boolean Algebra.
• Results of a boolean expression can be represented by a truth table.

Example: The expression \(Z = \overline A B + A \overline B\) has the following truth table:

• AND function
  • Theorem #1
    • \(X \cdot 0 = 0\)
  • Theorem #2
    • \(X \cdot 1 = X\)
  • Theorem #3
    • \(X \cdot X = X\)
  • Theorem #4
    • \(X \cdot \overline X = 0\)
• OR function
  • Theorem #5
    • \(X + 0 = X\)
  • Theorem #6
    • \(X + 1 = 1\)
  • Theorem #7
    • \(X + X = X\)
  • Theorem #8
    • \(X + \overline X = 1\)
• Invert Function
  • Theorem #9
    • \(\overline{\overline{X}} = X\)
• Communative Law
  • Theorem #11 (AND Function)
    • \(X \cdot Y = Y \cdot X\)
  • Theorem #12 (OR Function)
    • \(X + Y = Y + X\)
• Associative Law
  • Theorem #13 (AND Function)
    • \(X(YZ) = (XY)Z\)
  • Theorem #14 (OR Function)
    • \(X + (Y + Z) = (X + Y) + Z\)
• Distributive Law
  • Theorem #15 (AND Function)
    • \(X(X + Z) = XY + XZ\)
  • Theorem #16 (OR Function)
    • \((X + Y)(W + Z) = XW + XZ + YW + YZ\)
• Consensus Theorem
  • Theorem #16
    • \(X + \overline X Y = X + Y\)
  • Theorem #17
    • \(\overline X + XY + \overline X + Y\)
  • Theorem #18
    • \(X + \overline X \overline Y = X + \overline Y\)
  • Theorem #19
    • \(\overline X + X \overline Y = \overline X + \overline Y\)
• DeMorgan's Theorems
  • Theorem #20
    • \(\overline{XY} = \overline X + \overline Y\)
  • Theorem #21
    • \(\overline{X+Y} = \overline X \cdot \overline Y\)

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