Circuits, Truth Tables and Equations

• This is from Carpinelli chapter 2.
  • In L&S we will deal with this in more depth.
• Given an equation, we can "prove" it with a truth table.
  • This is a form of exhaustive proof which is possible because of the limited set of inputs.
• show ab = ba

  a	b	ab	ba
  0	0	0	0
  0	1	0	0
  1	0	0	0
  1	1	1	1

  • As a side a=b is the same as nxor

    • a	b	a=b	a⊕b	a⊕b
      0	0	1	0	1
      0	1	0	1	0
      1	0	0	1	0
      1	1	1	0	1

    • We can build a circuit to test this
    • 
  • So we could build a circuit to test our original property.
    • 
    • Note, when we test this circuit, it never turns off, which is what we want.
• I a math class (discrete I think) you will do more with boolean algebra
  • L&S will do more soon
• Let's look at the expression f(a,b) = ab
  • Can we form a circuit for this expression
    
  • Can we create the truth table.
    • We will do this by building a row for each of the simple terms of the expression.
      Iin this case, the only term is b

      a	b	b	ab
      0	0	1	0
      0	1	0	0
      1	0	1	1
      1	1	0	0

• Does a·b = a·b?
  • Build the truth table.
  • Build the circuit.
  • Note this is one part of DeMorgan's Laws.
• you can use an equation to produce a truth table or a circuit.
• You can also read an equation from a circuit.
• You can use a truth table to verify a circuit matches an equation.
• Or the other way around.
• Circuits for this section.