Boolean Algebra

• And will be written as a ⋅ (a⋅b)
• Or will be written as a + (a+b)
• Not has bar over the variable. (-a for my html)
• Some properties
  • a + 0 = a
  • a + 1 = 1
  • a ⋅ 0 = 0
  • a ⋅ 1 = a
  • a + -a = 1
  • a ⋅ -a = 0
  • a+b = b+a
  • a + (b + c) = (a + b) + c
  • a⋅(b+c) = a⋅b + a⋅c
  • Others on C-6
• We will occasionally use boolean algebra to simplify circuits.
• Represent x= a(b+c) + -a(b+c) as a circuit, truth table,
• A decoder
  • has n input lines
  • has 2ⁿ output lines
  • uses the bit pattern on the input lines to select a single line for output.
  • truth table on C-9
  • Build one
• An encoder
  • Has 2ⁿ input lines
  • Only one input line may contain data
  • n output lines
  • The address of the active input line is encoded on the output lines.
  • Build a truth table.
  • Build a circuit.
• A multiplexer
  • Has 2ⁿ input lines
  • Has n control lines
  • Has 1 output line
  • Based on the control lines, the proper input line is transferred to the output line.
• These circuits may be built to deal with "wide" input (more than 1 bit)
• An example