Introduction to Boolean Algebras

Objectives

1. Introduce boolean algebra
2. Introduce the basic operators.

Notes

• We are starting chapter 2 of Carpinelli
  • I don't know how much of this you get in discrete.
  • But this is tremendously important for the balance of the semester.
• The boolean algebra we will study has only two digits.
  • There are others, but this is the one we need.
  • And I will probably mistakenly refer to this as boolean algebra
• This was developed from Leibniz's (1646 - 1716) time (all praise Leibniz)
  • But mostly by and after George Bool (1815 - 1864)
  • In the 1938 Claude Shannon's Masters thesis implied that boolean algebra could be applied to building digital computers.
• Boolean algebra matches what we are going to do quite nicely
  • The one we will observe has two symbols, true and false
  • But we will generally represent these as 0 and 1
  • AND YOU WILL USE 0 and 1.
• There are a fixed set of operations.
• The nice part of the limited values is that we can explain and prove things by exhaustion.
  • This is the truth table.

  • input	output
    input values	output values

• For example, not is a binary function
  • one input
  • That input is inverted.

  • a	a
    0	1
    1	0

  • Note: since there was one input, there were 2¹=2 rows.
  • Note: I counted up from 0 to 1.
  • Note: I used a for not(a), NOT ¬a, a', ~a or anything else.
• For example Show a = a̿

  • a	a	a̿
    0	1	0
    1	0	1

• And is a two input operation
  • The symbol we will use is · or ab
  • Not ∧

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

  • Note we have 2 inputs so 2² = 4 rows.
  • Note we count from 0 to 4.
  • And indicates that both are a 1.
  • Note that 1 is the identity element.
    • 0 · 1 = 0
    • 1 · 1 = 1
    • so a · 1 = a
• Or is a two input operation.
  • The symbol we will use is + or a+b
  • Not ∨

  • a	b	a+b
    0	0	0
    0	1	1
    1	0	1
    1	1	1

  • Note that 0 is the identify element.
    • 0 + 0 = 0
    • 1 + 0 = 1
    • so a + 0 = a
  • This is the inclusive or. One or the other or both.
• There are other operations
  • nand : not and
  • nor : not or
  • xor : exclusive or
  • WE will examine these as needed.
• But why do we care about these?
  • Introduce digital.
  • Show input, output.
  • Show and, or, not.