Chapter 5 Boolean Algebra and Reduction Techniques 1

Figure 5.1 Combinational logic requirements for an automobile warning buzzer. Combinational logic uses two or more logic gates to perform a more useful, complex function. A combination of logic functions B = KD + HD Boolean Reduction B = D(K+H)

Figure 5.2 Reduced logic circuit for the automobile buzzer.

Discussion Point Write the Boolean equation for the circuit below: 6

5-2 Boolean Algebra Laws and Rules - Commutative laws Commutative laws of addition (A+B = B+ A) and multiplication (AB = BA) –The order of the variables does not matter. 7

Associative laws Associative laws of addition A + (B + C) = (A + B) + C and multiplication A(BC) = (AB)C The grouping of several variables Ored or ANDed together does not matter. 8

Distributive laws Distributive laws show methods for expanding an equation containing ORs and ANDs. A(B + C) = AB + AC (A + B)(C + D) = AC + AD + BC + BD 9

Boolean Laws and Rules Rule 1: Anything ANDed with a 0 equals 0 –A 0 = 0 Rule 2: Anything ANDed with a 1 equals itself –A 1 = A 10

Boolean Laws and Rules Rule 3: Anything ORed with a 0 equals itself –A + 0 = A Rule 4: Anything ORed with a 1 is equal to 1 –A + 1 = 1 11

Boolean Laws and Rules Rule 5: Anything ANDed with itself is equal to itself –A A = A Rule 6: Anything ORed with itself is equal to itself –A + A = A 12

Boolean Laws and Rules Rule 7: Anything ANDed with its complement equals 0 –A A = 0 Rule 8: Anything ORed with its complement equals 1 –A + A = 1 13

Boolean Laws and Rules Rule 9: Anything complemented twice will return to its original logic level –A = A 14

Boolean Laws and Rules Rule 10: –A + Ā B = A + B –Ā + AB = Ā + B 15

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5-3 Simplification of Combinational Logic Circuits Using Boolean Algebra Reduction of combinational logic circuits: equivalent circuits can be formed with fewer gates –Cost is reduced –Reliability is improved Approach: be performed by using laws and rules of Boolean Algebra 18

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