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CS 230 – Introduction to Computers and Computer Systems

Lecture 2 – Boolean Algebra and Circuits

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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.

Boolean OR

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Boolean OR

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Boolean OR

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Boolean OR

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Boolean AND

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Boolean NOT

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Boolean NAND

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¬ (X ∧ Y)

Boolean NOR

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¬ (X ∨ Y)

Boolean XOR

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(X ∧ ¬Y) ∨ (¬X ∧ Y)

Boolean XNOR

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(X ∧ Y) ∨ (¬X ∧ ¬Y)

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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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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

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

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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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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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