Text of Figure 2.6. A truth table for the AND and OR operations. 2.3 Truth Tables 1

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Figure 2.6. A truth table for the AND and OR operations.

2.3 Truth Tables1Figure 2.7. Three-input AND and OR operations.

2x 1 x 2 x n x 1 x 2 x n + + + x 1 x 2 x 1 x 2 + (b) OR gatesx 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 x n 2.4 Logic Gates and networks3Figure 2.9. The function from Figure 2.4.x 1 x 2 x 3 f x 1 x 2 + ( ) x 3 = 4S Power supplyS Light S X1X2X3An example of logic networks5x 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 x 1 x 2 f x 1 x 2 , ( ) 0 1 0 1 0 0 1 1 1 1 0 1 (b) Truth tableA B1 01 00 00 1Example (Cont): timing diagram61 0 1 0 1 0 1 0 1 0 x 1 x 2 A B f Time(c) Timing diagramx 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/O71 1 0 0 0 0 1 1 1 1 0 1 0 1 0 1 g x 1 x 2 2.5 Boolean Algebra foundation for modern digital technologyIn 1849, first published by George Boole for the algebraic description of processes involved in logical thought and reasoning.In late 1930s, 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.8Axioms of Boolean algebra9Single-variable theorems10If x is a Boolean variable10Principle of dualityGiven a logic expression, its dual is obtainedby 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 algebraOften, one expression leads to a simpler physical implementation.11Axioms and theorems listed are in pairs 11DeMorgans Theorem12Two- and Three- Variable propertiescommutative10a x y = y x 10b x + y = y + xAssociative11a x (y z) = (x y) z11b x + (y + z) = (x + y) + zDistributive12a x (y + z) = x y + x z12b x + y z = (x + y) (x + z)Absorption13a x + x y = x13b x (x+y) = x