Nov 20, 2012
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IT2001PAEngineering Essentials (2/2)
Chapter 8 - Applications of Boolean Algebra
2
Chapter 8 - Applications of Boolean Algebra
IT2001PA Engineering Essentials (2/2)
Lesson Objectives
Upon completion of this topic, you should be able to:
Apply Boolean algebra to solve combination logic of up to 2 variables.
Chapter 8 - Applications of Boolean Algebra
IT2001PA Engineering Essentials (2/2)
Specific Objectives
Students should be able to : Determine the output logic expression from a given
logic circuit.
Use Boolean algebra to simplify logic expressions.
4
Chapter 8 - Applications of Boolean Algebra
IT2001PA Engineering Essentials (2/2)
Example 1
Write the Boolean equation for the circuit
Use DeMorgan’s Theorem and then Boolean Algebra rules to simplify the equation. Draw the simplified circuit.
5
Chapter 8 - Applications of Boolean Algebra
IT2001PA Engineering Essentials (2/2)
Example 1 (Solution)
Boolean equation at X, X = (AB)●B
Apply De Morgan’s theorem:
X = (A + B) ● B(Use parentheses, () to maintain proper grouping)
Apply Distributive Law:
X = AB + BB
Apply Boolean Algebra rule:
= AB + 0
= AB
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Chapter 8 - Applications of Boolean Algebra
IT2001PA Engineering Essentials (2/2)
Example 1 (Solution)
The simplified equation will be
X=AB
To draw the simplified circuit, first write down the inputs (A and B), and the output (X),
then insert the corresponding gates
Finally, join the gates.
7
Chapter 8 - Applications of Boolean Algebra
IT2001PA Engineering Essentials (2/2)
Example 2
Write the Boolean equation for the circuit.
Use DeMorgan’s Theorem and then Boolean Algebra rules to simplify the equation. Draw the simplified circuit.
8
Chapter 8 - Applications of Boolean Algebra
IT2001PA Engineering Essentials (2/2)
Example 2 (Solution)
X = B●(A+C)+C, Simplifying equation:Boolean Equation at X: X = B●(A + C) + C
Apply Distributive Law: X = BA + BC + C
Since C is common for term 2 and term 3: X = BA + C●(B + 1)
Apply (B + 1) = 1: X = BA + C● 1
Apply C●1 = C: X = BA + C
Apply BA = AB: X = AB + C
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Chapter 8 - Applications of Boolean Algebra
IT2001PA Engineering Essentials (2/2)
Example 2 (Solution)
The simplified equation will be:
X = AB + C
To draw the simplified circuit, first write down the inputs (A, B and C),and the output (X),
then insert the corresponding gates.
Finally, join the gates.
10
Chapter 8 - Applications of Boolean Algebra
IT2001PA Engineering Essentials (2/2)
Example 3
Write the Boolean equation for the circuit.
Use DeMorgan’s Theorem and then Boolean Algebra rules to simplify the equation. Draw the simplified circuit.
11
Chapter 8 - Applications of Boolean Algebra
IT2001PA Engineering Essentials (2/2)
Example 3 (Solution)
X = (A+B)●BC+A Simplifying equation:
Boolean Equation at X: X = (A+B) ● BC + A
Apply Distributive Law: X = ABC + BBC + A
Apply BBC = BC: X = ABC + BC+ A
Since BC is common for term 1 and term 2: X = BC (A +1) + A
Apply (A + 1) = 1: X = BC + A
12
Chapter 8 - Applications of Boolean Algebra
IT2001PA Engineering Essentials (2/2)
Example 3 (Solution)
The simplified equation will be:
X = BC + A
To draw the simplified circuit, first write down the inputs (A, B and C),and the output (X),
then insert the corresponding gates.
Finally, join the gates.
13
Chapter 8 - Applications of Boolean Algebra
IT2001PA Engineering Essentials (2/2)
Example 4
Write the Boolean equation for the circuit.
Use DeMorgan’s Theorem and then Boolean Algebra rules to simplify the equation. Draw the simplified circuit.
14
Chapter 8 - Applications of Boolean Algebra
IT2001PA Engineering Essentials (2/2)
Example 4 (Solution)
X = (A+B)●B+(B)+(BC)
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Chapter 8 - Applications of Boolean Algebra
IT2001PA Engineering Essentials (2/2)
Example 4 (Solution)
Simplifying equation:
Boolean Equation at X: X = (A + B)●B + B + BC
Apply Distributive Law: X = AB + BB + B + BC
Apply (BB = 0): X = AB + 0 + B + BC
Apply (AB + 0) = AB: X = AB + B + BC
Since B is common for term 1 and term 2:X = B●(A + 1) + BC
Apply (A + 1) = 1: X = B ● 1 + BC
Apply (B●1) = B: X = B + BC
Apply (B +BC) = B+C: X = B + C
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Chapter 8 - Applications of Boolean Algebra
IT2001PA Engineering Essentials (2/2)
Example 4 (Solution) The simplified equation will be:
X = B + A
To draw the simplified circuit, first write down the inputs (A, B and C),and the output (X),
then insert the corresponding gates.
Finally, join the gates.
17
Chapter 8 - Applications of Boolean Algebra
IT2001PA Engineering Essentials (2/2)
Example 5
Write the Boolean equation for the circuit.
Use DeMorgan’s Theorem and then Boolean Algebra rules to simplify the equation. Draw the simplified circuit.
18
Chapter 8 - Applications of Boolean Algebra
IT2001PA Engineering Essentials (2/2)
Example 5 (Solution)
X = AB●(C+D)●AB
Simplifying Equation:
Boolean Equation at X: X= AB●(C+D)●AB
Apply Demorgan’s Thereom: X= AB + (C+D) + AB
Apply C+D = C+D: X= AB + C+D + AB
Apply AB + AB = AB: X= AB + C+D
Apply AB = A + B: X= A + B + C+D
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Chapter 8 - Applications of Boolean Algebra
IT2001PA Engineering Essentials (2/2)
Example 5 (Solution)
The simplified equation will be:
X = A + B + C + D
To draw the simplified circuit, first write down the inputs (A, B,C and D), and the output (X),
then insert the corresponding gates.
Finally, join the gates.
20
Chapter 8 - Applications of Boolean Algebra
IT2001PA Engineering Essentials (2/2)
Summary
Steps taken to simplify a combination logic circuit: Write down the expression of a given Combination Logic
Circuit.
Simplify the expression using Boolean Algebra Theorem.
Draw the Logic Circuit using the simplified Logic Expression.
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Chapter 8 - Applications of Boolean Algebra
IT2001PA Engineering Essentials (2/2)
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