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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

  1. CS 230 – Introduction to Computers and Computer Systems Lecture 2 – Boolean Algebra and Circuits CS 230 - Spring 2020 1-1

  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. CS 230 - Spring 2020 1-2

  3. Boolean OR CS 230 - Spring 2020 1-3

  4. Boolean OR CS 230 - Spring 2020 1-4

  5. Boolean OR CS 230 - Spring 2020 1-5

  6. Boolean OR CS 230 - Spring 2020 1-6

  7. Boolean AND CS 230 - Spring 2020 1-7

  8. Boolean NOT CS 230 - Spring 2020 1-8

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  23. Boolean NAND ¬ (X ∧ Y) CS 230 - Spring 2020 1-23

  24. Boolean NOR ¬ (X ∨ Y) CS 230 - Spring 2020 1-24

  25. Boolean XOR (X ∧ ¬Y) ∨ (¬X ∧ Y) CS 230 - Spring 2020 1-25

  26. Boolean XNOR (X ∧ Y) ∨ (¬X ∧ ¬Y) CS 230 - Spring 2020 1-26

  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 CS 230 - Spring 2020 1-27

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  53. Boolean Algebra Rules  Identities  Annihilators A ∨ 0 = A A ∨ 1 = 1 A ∧ 1 = A A ∧ 0 = 0 A ∨ A = A  Complements A ∧ A = A A ∨ ¬A = 1 A ∧ ¬A = 0  Involution ¬¬A = A CS 230 - Spring 2020 1-53

  54. 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

  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 CS 230 - Spring 2020 1-55

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  76. Digital Electronic Circuits  Boolean algebra models electronic circuits  Each operation has a corresponding gate  A physical circuit component that applies a Boolean operator AND NAND XOR OR NOR XNOR NOT CS 230 - Spring 2020 1-76

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