Lecture 2 Boolean Algebra and Circuits CS 230 - Spring 2020 1-1 - - PowerPoint PPT Presentation

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Lecture 2 Boolean Algebra and Circuits CS 230 - Spring 2020 1-1 - - PowerPoint PPT Presentation

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CS 230 Introduction to Computers and Computer Systems Lecture 2 Boolean Algebra and Circuits CS 230 - Spring 2020 1-1 Boolean Algebra Algebra to express binary logic Basic operators: OR, AND, NOT More advanced operators later

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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.

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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)

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Boolean NOR

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¬ (X ∨ Y)

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Boolean XOR

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(X ∧ ¬Y) ∨ (¬X ∧ Y)

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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

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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

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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

  • perator
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