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
Boolean Reduction
B = KD + HD
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 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
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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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