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CS 230 - Spring 2020 1-1
CS 230 – Introduction to Computers and Computer Systems
Lecture 2 – Boolean Algebra and Circuits
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CS 230 - Spring 2020 1-2
Boolean Algebra
Algebra to express binary logic
Basic operators: OR, AND, NOT
More advanced operators later
Like normal elementary algebra
OR – similar to addition
AND – similar to multiplication
NOT – similar to negation
identity, inverse, commutative, associative, etc.
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Boolean OR
CS 230 - Spring 2020 1-3
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Boolean OR
CS 230 - Spring 2020 1-4
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Boolean OR
CS 230 - Spring 2020 1-5
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Boolean OR
CS 230 - Spring 2020 1-6
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Boolean AND
CS 230 - Spring 2020 1-7
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Boolean NOT
CS 230 - Spring 2020 1-8
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CS 230 - Spring 2020 1-9
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CS 230 - Spring 2020 1-10
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CS 230 - Spring 2020 1-11
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CS 230 - Spring 2020 1-12
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CS 230 - Spring 2020 1-13
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CS 230 - Spring 2020 1-14
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CS 230 - Spring 2020 1-15
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CS 230 - Spring 2020 1-16
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CS 230 - Spring 2020 1-17
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CS 230 - Spring 2020 1-18
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CS 230 - Spring 2020 1-19
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CS 230 - Spring 2020 1-20
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CS 230 - Spring 2020 1-21
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CS 230 - Spring 2020 1-22
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Boolean NAND
CS 230 - Spring 2020 1-23
¬ (X ∧ Y)
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Boolean NOR
CS 230 - Spring 2020 1-24
¬ (X ∨ Y)
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Boolean XOR
CS 230 - Spring 2020 1-25
(X ∧ ¬Y) ∨ (¬X ∧ Y)
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Boolean XNOR
CS 230 - Spring 2020 1-26
(X ∧ Y) ∨ (¬X ∧ ¬Y)
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CS 230 - Spring 2020 1-27
Precedence
Boolean operators have precedence
Just like regular algebra
Order is (from highest to lowest)
NOT > AND=NAND > XOR=XNOR > OR=NOR
Equal precedence are evaluated left-to-right
Parenthesis override precedence
Just like regular algebra
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CS 230 - Spring 2020 1-28
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CS 230 - Spring 2020 1-29
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CS 230 - Spring 2020 1-30
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CS 230 - Spring 2020 1-31
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CS 230 - Spring 2020 1-32
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CS 230 - Spring 2020 1-33
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CS 230 - Spring 2020 1-34
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CS 230 - Spring 2020 1-35
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CS 230 - Spring 2020 1-36
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CS 230 - Spring 2020 1-37
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CS 230 - Spring 2020 1-38
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CS 230 - Spring 2020 1-39
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CS 230 - Spring 2020 1-40
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CS 230 - Spring 2020 1-41
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CS 230 - Spring 2020 1-42
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CS 230 - Spring 2020 1-43
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CS 230 - Spring 2020 1-44
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CS 230 - Spring 2020 1-45
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CS 230 - Spring 2020 1-46
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CS 230 - Spring 2020 1-47
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CS 230 - Spring 2020 1-48
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CS 230 - Spring 2020 1-49
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CS 230 - Spring 2020 1-50
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CS 230 - Spring 2020 1-51
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CS 230 - Spring 2020 1-52
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CS 230 - Spring 2020 1-53
Boolean Algebra Rules
Identities
A ∨ 0 = A
A ∧ 1 = A
A ∨ A = A
A ∧ A = A
Involution
¬¬A = A
Annihilators
A ∨ 1 = 1
A ∧ 0 = 0
Complements
A ∨ ¬A = 1
A ∧ ¬A = 0
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More Rules
Commutative Law
A ∨ B = B ∨ A
A ∧ B = B ∧ A
Associative Law
A ∨ (B ∨ C) = (A ∨ B) ∨ C
A ∧ (B ∧ C) = (A ∧ B) ∧ C
CS 230 - Spring 2020 1-54
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CS 230 - Spring 2020 1-55
Even More Rules
Distributive Law
A ∧ (B ∨ C) = A ∧ B ∨ A ∧ C
A ∨ (B ∧ C) = (A ∨ B) ∧ (A ∨ C)
De Morgan's Law
¬(A ∨ B) = ¬A ∧ ¬B
¬(A ∧ B) = ¬A ∨ ¬B
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CS 230 - Spring 2020 1-56
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CS 230 - Spring 2020 1-57
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CS 230 - Spring 2020 1-58
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CS 230 - Spring 2020 1-59
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CS 230 - Spring 2020 1-60
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CS 230 - Spring 2020 1-61
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CS 230 - Spring 2020 1-62
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CS 230 - Spring 2020 1-63
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CS 230 - Spring 2020 1-64
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CS 230 - Spring 2020 1-65
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CS 230 - Spring 2020 1-66
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CS 230 - Spring 2020 1-67
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CS 230 - Spring 2020 1-68
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CS 230 - Spring 2020 1-69
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CS 230 - Spring 2020 1-70
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CS 230 - Spring 2020 1-71
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CS 230 - Spring 2020 1-72
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AND NAND XOR
OR NOR XNOR
NOT
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Digital Electronic Circuits
Boolean algebra models electronic circuits
Each operation has a corresponding gate
A physical circuit component that applies a Boolean operator
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