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  x n +++ x 1 x 2 x 1 x 2 + (b) OR gates x x (c) NOT gate Figure 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  x n 2.4 Logic Gates and networks 3

Figure 2.9. The function from Figure 2.4. x 1 x 2 x 3 fx 1 x 2 +  x 3  = 4 S Power supplyS Light S X1X1 X2X2 X3X3

An example of logic networks 5 x 1 x  f 0001  1101  0011  0101  A B x 1 x 2 fx 1 x 2,() (b) Truth table A B

Example (Cont’): timing diagram x 1 x 2 A B f Time (c) Timing diagram x 1 x  f 0001  1101  0011  0101  A B

Example (Cont’): another network with same logic behavior at I/O  0011  1101  0101  g x 1 x 2

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 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 12 xyx+y

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 13