Figure 2.6. A truth table for the AND and OR operations.
2.3 Truth Tables
1
Figure 2.7. Three-input AND and OR operations.
2
x 1 x 2
x n
x 1 x 2 ¼ x n + + + x 1 x 2
x 1 x 2 +
(b) OR gates
x x
(c) NOT gateFigure 2.8. The basic gates.
(a) AND gates
x 1 x 2
x n
x 1 x 2
x 1 x 2 x 1 x 2 ¼ x n
2.4 Logic Gates and networks
3
Figure 2.9. The function from Figure 2.4.
x 1 x 2 x 3
f x 1 x 2 + ( ) x 3 × =
4
S
Power supply S
Light
S
X1
X2
X3
An example of logic networks
5
x 1
x 2
1 1 0 0 ® ® ®
f 0 0 0 1 ® ® ®
1 1 0 1 ® ® ®
0 0 1 1 ® ® ®
0 1 0 1 ® ® ®
(a) Network that implements
A
B
x 1 x
2 f x 1 x
2 , ( )
0 1 0 1
0 0 1 1
1 1 0 1
(b) Truth table
A B
1 0
1 0
0 0
0 1
Example (Cont’): timing diagram
6
1 0
1 0
1 0
1 0
1 0
x 1
x 2
A
B
f Time
(c) Timing diagram
x 1
x 2
1 1 0 0 ® ® ®
f 0 0 0 1 ® ® ®
1 1 0 1 ® ® ®
0 0 1 1 ® ® ®
0 1 0 1 ® ® ®
A
B
Example (Cont’): another network with same logic behavior at I/O
7
1 1 0 0 ® ® ® 0 0 1 1 ® ® ®
1 1 0 1 ® ® ® 0 1 0 1 ® ® ® g
x 1
x 2
(d) Network that implements
2.5 Boolean Algebra – foundation for modern digital technology
• In 1849, first published by George Boole for the algebraic description of processes involved in logical thought and reasoning.
• In late 1930’s, Claude Shannon show that Boolean algebra provides an effective means of describing circuits built with switches.– -> Algebra is a powerful tool for designing and
analyzing logic circuits.
8
Axioms of Boolean algebra
1a. 0 ∙ 0 = 0
1b. 1 + 1 = 1
2a. 1 ∙ 1 = 1
2b. 0 + 0 = 0
3a. 0 ∙ 1 = 1 ∙ 0 = 0
3b. 1 + 0 = 0 + 1 = 1
4a. If x = 0, then
4b. If x = 1, then = 0
9
Single-variable theorems
•
10
Principle of duality
• Given a logic expression, its dual is obtained– by replacing all + operators with ∙ operators, and vice
versa.– By replacing all 0s with 1s, and vice versa.
• The dual of any true statement (axioms or theorems) in Boolean algebra is also true.– Later on, we will show that duality implies that at least
two different ways exist to express every logic function with Boolean algebra
• Often, one expression leads to a simpler physical implementation.
11
DeMorgan’s Theorem
x + y = x + y = x + y
12
x y + x+y x+y0 0 1 1 0 00 1 1 1 1 11 0 1 1 1 11 1 0 0 1 1
Two- and Three- Variable properties
commutative
10a x ∙ y = y ∙ x
10b x + y = y + x
Associative
11a x ∙ (y ∙ z) = (x ∙ y) ∙ z
11b x + (y + z) = (x + y) + z
Distributive
12a x ∙ (y + z) = x ∙ y + x ∙ z
12b x + y ∙ z = (x + y) ∙ (x + z)
Absorption
13a x + x ∙ y = x
13b x ∙ (x+y) = x
Combining
14a x ∙ y + x ∙
14b (x + y) ∙ (x + ) = x
DeMorgan’s Theorem15a
15b 16a x + y = x + y
16b = x y
Consensus
17a x y + z + y z = x y + z
17b (x+y)()(y+z)=(x+y)()
13
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