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Chapter 5

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

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Chapter 5

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  1. Chapter 5 Boolean Algebra and Reduction Techniques 1

  2. 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)

  3. Figure 5.2 Reduced logic circuit for the automobile buzzer.

  4. Discussion Point • Write the Boolean equation for the circuit below: 6

  5. 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

  6. 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

  7. 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

  8. 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

  9. 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

  10. 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

  11. 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

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

  13. Boolean Laws and Rules • Rule 10: • A + Ā B = A + B • Ā + AB = Ā + B 15

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  15. 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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