Boolean Expressions

• I am using Carpinelli chapter 2.
• We are looking at section 2 of the outline.
  • 2 a : Combinational logic is a circuit where the output is completely determined by the input.
  • Sequential logic, is a circuit which has a state and the output is determined by a combination of the input and that state.
    • This is a circuit with a memory.
  • Everything we have looked at so far is combinational.
  • We will look at each of these in the future.
• We have covered 2b and 2c
• Another warning:
  • Truth tables will be built with 0 and 1 not T/F
  • Rows in truth tables will be ordered from 0 to $2^n-1$ for n inputs.
  • You will use the logical operators +,· and  . If you need ' will do for not.
• Boolean Properties.
  • Closure: all of the boolean operators will produce a value of either 0 or 1.
    • This is like addition under the integers, add any two integers and you get another integer.
    • Division is not closed under the integers. 1/2 is not an integer.
  • There exists an identity element
    • In arithmetic:
      • a + 0 = a
      • 0 is the additive identity
      • a × 1 = a
      • 1 is the multiplicative identity.
    • What is the identity for or? 0
      • 0 + 0 = 0
      • 1 + 0 = 1
    • What is the identity for and? 1
      • 0 · 1 = 0
      • 1 · 1 = 1
  • There is an inverse
    • the inverse of a is a
    • a · a = 0
    • a + a = 1
  • Commutative property
    • ab = ba, a+b = b+a

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

  • Associative properties
    • (a+b)+c = a+(b+c)

      a	b	c	a+b	(a+b)+c	b+c	a+(b+c)
      0	0	0	0	0	0	0
      0	0	1	0	1	1	1
      0	1	0	1	1	1	1
      0	1	1	1	1	1	1
      1	0	0	1	1	0	1
      1	0	1	1	1	1	1
      1	1	0	1	1	1	1
      1	1	1	1	1	1	1

    • (ab)c = a(bc)
    • Table is similar
  • Distributive properties
    • a(b+c) = ab+ac

      a	b	c	b+c	a(b+c)	ab	ac	ab+ac
      0	0	0	0	0	0	0	0
      0	0	1	1	0	0	0	0
      0	1	0	1	0	0	0	0
      0	1	1	1	0	0	0	0
      1	0	0	0	0	0	0	0
      1	0	1	1	1	0	1	1
      1	1	0	1	1	1	0	1
      1	1	1	1	1	1	1	1

• The following are helpful when simplifying circuits
  • Idempotence
    • When a function takes the same input twice, it is the value of the input.
    • This is helpful when simplifying circuits.

      a	aa	a+a
      0	0	0
      1	1	1

  • Involution
    • The inverse of the inverse is the item

      a	a	a̿
      0	1	0
      1	0	1

    • I used a unicode trick for the double bar.
      • a̿ is a followed by the unicode character 831 (in decimal)
  • Absorption
    • a(a+b) = a, a+(ab) = a

      a	b	a+b	a(a+b)	ab	a+(ab)
      0	0	0	0	0	0
      0	1	1	0	0	0
      1	0	1	1	0	1
      1	1	1	1	1	1

• De Morgan's
  • a+b = a·b
  • ab = a+b