Chapter 5

Boolean Algebra and Reduction Techniques

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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 + HDBoolean 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:

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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.

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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.

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Distributive lawsDistributive laws show methods for expanding an equation containing ORs and ANDs.A(B + C) = AB + AC

(A + B)(C + D) = AC + AD + BC + BD

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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

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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

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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

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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

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Boolean Laws and Rules

• Rule 9: Anything complemented twice will return to its original logic level– A = A

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Boolean Laws and Rules• Rule 10:

– A + Ā B = A + B– Ā + AB = Ā + B

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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

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