Boolean Logic and Gates

1
Boolean Logic and Gates

• CS227
• Western Washington University

2
Boolean Algebra

• defined with a set of elements, a set of
  operators, and a number of postulates or axioms.
• set of elements is 0 , 1
• set of operators is ,
• 2-valued boolean algebra is also called binary
  logic or switching algebra
• Order of operator precedence is
• ( ), NOT, AND, OR

3
Huntington Postulates

• 1. a. Closure wrt operater
• b. Closure wrt operator
• 2. a. Identity element wrt operator, designated
  by 0 x 0 x
• b. Identity element wrt operator,
  designated by 1 x 1 x
• 3. a. Commutative wrt operator x y y x
• b. Commutative wrt operator x y y
  x
• 4. a. is distributive over x (y z)
  (x y) (x z)
• b. is distributive over x (y z)
  (x y) (x z)
• 5. For every element x, there exists an element
  x, called the complement, such that x x 1
  and x x 0
• 6. There exists at least 2 elements x y, such
  that x ! y

4
Basic Theorems

• Involution
• (x) x
• Associative
• x (y z) (x y) z
• x (y z) (x y) z
• DeMorgan
• (x y) x y
• (x y) x y
• Absorption
• x (x y) x
• x (x y) x

5
The Device Level
The device level is actually a level below what
weve been referring to as level 0.
Understanding the device level requires some
understanding of electrical engineering. At this
level, the fundamental elements are gates. Gates
are made up of transistors. At this level,
only 2 logical values are present. The logical
values are determined by the voltage of a
signalusually after passing through a
transistor.
6
Positive vs. Negative Logic
Since the logic value depends on the voltage of a
signal, we have 2 options for representing
logical 1 and logical 0. Positive logic
represents logical 0 as 0 volts and logical 1 as
3.3 or 5 volts. Negative logic represents
logical 0 as 3.3 or 5 volts and logical 1 as 0
volts.
7
Logic Gate Symbols
8
Boolean Functions

• can be represented as algebraic expressions
• F xyz
• can be represented in a truth table
• can be transformed into a logic diagram composed
  of AND, OR, and NOT gates
• each literal (variable) designates an input to a
  gate
• each term is implemented with a logic gate
• why would we want to minimize the number of
  literals and the number of terms?
• x xy
• x(x y)