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04 - Boolean Algebra and Circuits
lecture notes
Course
Introduction to Computer Systems (Comp 273)
80 Documents
Students shared 80 documents in this course
University
McGill University
Academic year: 2020/2021
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Boolean Algebra and Circuits
Overview
á Truth Tables
á Boolean Functions
á Logic Gates
á Combinatorial Circuits
á Appendix C
Boolean Operations
á Consider binary variables, A, B
± Possible values are 0 (FALSE) and 1 (TRUE)
á Different possible inputs with n operands = 2n
± 2 binary operands: 2 2 = 4 different possible inputs
± 4 binary operands: 2 4 = 16 different possible inputs
A A B
0
1
A B C
Boolean Operations
á Consider binary variables, A, B
± Possible values are 0 (FALSE) and 1 (TRUE)
á Different possible inputs with n operands = 2n
± 2 binary operands: 2 2 = 4 different possible inputs
± 4 binary operands: 2 4 = 16 different possible inputs
A B
0 0
0 1
1 0
1 1
A
0
1
A B C
Ï 2 Ï 2 Ï 2
Ï 2 Ï 2 Ì 2
Ï 2 Ì 2 Ï 2
Ï 2 Ì 2 Ì 2
Ì 2 Ï 2 2 Ï
Ì 2 Ï 2 Ì 2
Ì 2 Ì 2 Ï 2
Ì 2 Ì 2 Ì 2
Boolean Operations
á A Boolean Function is an algebraic expression consisting of
binary variables and the logic operation symbols
á Any Boolean function can be written in terms
of AND, OR, and NOT
á Truth table defines all possible outcomes A B | A + B
0 0 0 0
0 1 0 1
1 0 0 1
1 1 1 1
Boolean Algebra
á Rules capture logical reasoning
á Allow for manipulation of expressions
Are two boolean functions equivalent?ent?
Prove it, or find values to show that they are not!
||[=||[=||[ [
|~|[=|[
Boolean Algebra
Derive Boolean Expression
á Consider a logic function with three inputs, A , B , and C , and
three outputs, D , E , and F.
± D is true if at least one input is true
± E is true if exactly two inputs are true
± F is true only if all three inputs are true
á Find the Boolean expression for D, E, and F.
Derive Boolean Expression
Find the truth table! D is true if at least one input is true
A B C D
Ï 2 Ï 2 Ï 2
Ï 2 Ï 2 Ì 2
Ï 2 Ì 2 Ï 2
Ï 2 Ì 2 Ì 2
Ì 2 Ï2 2Ï
Ì 2 Ï 2 Ì 2
Ì 2 Ì 2 Ï 2
Ì 2 Ì 2 Ì 2
Derive Boolean Expression
Find the truth table! F is true only if all three inputs are true
A B C D E F
Ï 2 Ï 2 Ï 2 0 0
Ï 2 Ï 2 Ì 2 1 0
Ï 2 Ì 2 Ï 2 1 0
Ï 2 Ì 2 Ì 2 1 1
Ì 2 Ï2 2Ï 1 0
Ì 2 Ï 2 Ì 2 1 1
Ì 2 Ì 2 Ï 2 1 1
Ì 2 Ì 2 Ì 2 1 0
Derive Boolean Expression
What are the Boolean expressions for outputs D, E, and F?
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04 - Boolean Algebra and Circuits
Course: Introduction to Computer Systems (Comp 273)
80 Documents
Students shared 80 documents in this course
University: McGill University
Was this document helpful?
Boolean Algebra and Circuits
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