IS 151 Lecture 5 - UDSM 2013

1. Boolean Analysis of Logic
Circuits
Consider the following logic circuit
Determine the Boolean expression for the circuit
Construct the Truth Table for the circuit
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2. Boolean Analysis of Logic
Circuits
• Boolean expression:
– Assume output is X, then
• X = A(B + CD)
• The truth table
– Truth table shows the output of the logic
circuit for all possible values of the input
variables
– Input variables are: A, B, C and D
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3. Boolean Analysis of Logic
Circuits – Truth Table
Inputs
Output
A
B
C
D
A(B + CD)
0
0
0
0
0
0
0
0
1
0
0
0
1
0
0
0
0
1
1
0
0
1
0
0
0
0
1
0
1
0
0
1
1
0
0
0
1
1
1
0
1
0
0
0
0
1
0
0
1
0
1
0
1
0
0
1
0
1
1
1
1
1
0
0
1
1
1
0
1
1
1
1
1
0
1
1
1
1
1
1
Find the values of the variables that make the
expression equal to 1
• A(B + CD) = 1 only if A = 1 and (B + CD) = 1
• Determine when B + CD = 1
• B + CD = 1 if either B = 1 or CD = 1, or both
B and CD = 1
• CD = 1 only if C = 1 and D = 1
• Therefore, A(B + CD) = 1 when either:
• A = 1 and B = 1 regardless of values of C and
D, or
• A = 1, C = 1 and D = 1 regardless of the value
of B
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4. Boolean Analysis of Logic
Circuits – Truth Table
• Truth Table – Summary:
– A(B + CD) = AB + ACD
• Determine when AB + ACD = 1
– AB = 1 only when A = 1 AND B = 1 (regardless of C and
D)  Condition 1
– ACD = 1 only when A = 1, C = 1, AND D = 1, (regardless
of B)  Condition 2
• Wherever A = 1 AND B = 1, put a 1
• Wherever A = 1, C = 1 and D = 1, put a 1
• Otherwise put a 0
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5. Boolean Analysis of Logic
Circuits - Exercise
• In the previous example, replace AND
gates with OR gates and OR gates with
AND gates and :
– Determine the Boolean expression for the
output
– Construct a truth table for the circuit
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6. Standard Forms of Boolean
Expressions
• All Boolean expressions can be converted
into
– The Sum-of-Products form (SoP)
– The Product-of-Sum form (PoS)
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7. Standard Forms of Boolean
Expressions
• The Sum-of-Products form
– When two or more terms are summed by Boolean
addition, the resulting expression is a Sum-ofProducts (SOP)
– Examples:
• AB + ABC
• ABC + CDE + B’CD’
• A’B + A’BC’ + AC
– In an SOP expression, a single over bar cannot
extend over more than one variable
• E.g. (ABC)’ – need to be simplified
• A’B’C – already simplified
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8. Standard Forms of Boolean
Expressions
• The Sum-of-Products form – example
– AB + BCD + AC – show the expression
diagrammatically
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9. Standard Forms of Boolean
Expressions
• Converting a general expression to SOP form
– E.g. A(B + CD) = AB + ACD
• Exercise: Convert the following into SOP form
1. AB + B(CD + EF)
= AB + BCD + BEF
2. (A + B)(B + C + D)
= AB + AC + AD + BB + BC + BD
3. ((A + B)’ + C)’
= (A + B)’’.C’
= (A + B).C’
= AC’ + BC’
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10. Standard Forms of Boolean
Expressions
• The Product-of-Sum form:
– When two or more sum terms are multiplied,
the resulting expression is a Product-of-Sum
(POS).
– Examples:
• (A’ + B)(A + B’ + C)
• (A’ + B’ + C’)(C + D’ E)(B’ + C + D)
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11. Standard Forms of Boolean
Expressions
• The Product-of-Sum form – example
– (A + B)(B + C + D)(A + C) - show the
expression diagrammatically
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12. • End of lecture
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