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Boolean Algebra

Boolean Algebra. Topics. Binary Number System AND, OR, NOT, Exclusive OR, and Implications operations Truth Tables and Algebraic Laws Boolean Logic and Circuits Application in Internet Searching Boolean Logic and Fallacies. Binary. OFF 0. ON 1. Counting Binary. AND.

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Boolean Algebra

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  1. Boolean Algebra

  2. Topics • Binary Number System • AND, OR, NOT, Exclusive OR, and Implications operations • Truth Tables and Algebraic Laws • Boolean Logic and Circuits • Application in Internet Searching • Boolean Logic and Fallacies

  3. Binary OFF 0 ON 1

  4. Counting Binary

  5. AND Usually denoted with ^ Example: 1 ^ 1 = 1 1 ^ 0 = 0

  6. OR Usually denoted with ∨ Example: 1 ∨ 1 = 1 0 ∨ 0 = 0

  7. Example 1 • For x = 1 and y = 0 • (x ^ y ) ∨ (x ^ y) • (1 ^ 0) ∨ (1 ^ 0) • 0 ∨ 0 • 0

  8. NOT Usually denoted with ~ Example: ~1 = 0 ~0 = 1

  9. EXCLUSIVE OR Usually denoted with ⊕ Example: 1 ⊕ 1 = 0 0 ⊕ 1 = 1 c

  10. Example 2 • For x = 1, y = 1, z = 0 • ~(x ^ z) ⊕ ( y ∨ ~z) • ~(1 ^ 0) ⊕ ( 1 ∨ ~0) • ~0 ⊕ ( 1 ∨ 1) • 1 ⊕ 1 • 0

  11. Implication Usually denoted with => Example: 1 => 1 = 1 0 => 1 = 1

  12. Truth Tables • Table that lists ALL possible values for a particular equation. • Ex: (x ^ y ) ∨ (x ∨ y)

  13. Algebraic Laws • Associativity => x ∨ (y ∨ z) = (x ∨ y) ∨ z • Commutativity => x ∧ y = y ∧ x • Distribution => x ∧ ( y ∨ z ) = (x ∧ y) ∨ (x ∧ z) • De Morgan => (~x) ^ (y) = ~(x ∨ y)

  14. Application 1: Circuits

  15. Transistors and Gates • Gates – A single entity that usually take in two inputs and produces an output. • Transistors – Works the same way as logical gates, but is an actual device. Building blocks a computer. • Number of Transistors in a modern CPU: 10 Million

  16. Gates

  17. Example: Alarm

  18. Example: Half Adder • Adds 2 bits • 0 + 0 = 0 • 0 + 1 = 1 • 1 + 0 = 1 • 1 + 1 = 0 + 1 (carry_ • 0 + 0 + 1 = 1 • … • …

  19. Application 2: Searching

  20. Searching Primitives • “pizza restaurants” “Mexican restaurants” • Both pizza restaurants and Mexican restaurants must appear as your result. • “respiration” OR “photosynthesis” • At least one of these results must appear as your result. This can mean only respiration, both, or only photosynthesis.

  21. Searching Primitives • “William Shakespeare” -hamlet • Search for anything William Shakespeare related that does not include hamlet. Analogues to the NOT operator. • “Natural Disasters” +earthquakes • Search for anything that involves natural disasters, but the term earth quakes must be included.

  22. Example: Searching Techniques • Suppose I am interested in scholarships, but I don’t want to search for any women or grade based scholarships. • Scholarships AND NOT minority OR need based • scholarships –minority OR –grades

  23. Example 2: Searching Techniques • Let’s search for graduate school that are NOT in Alabama, but are in Georgia OR Florida • graduate schools -"Alabama" +"Florida" OR +"Georgia"

  24. Application 3: Analyzing Fallacies

  25. Fallacy I • A caused B therefore B caused A • Ex: Bill sets out several plates with bread on them. After a couple days, he notices that the bread has mold growing all over it. Bill concludes that the mold was produced by the bread going bad. • A => B must be equal to B => A. Let’s test that out!

  26. Fallacy II • Is this a fallacy? • A cause B therefore B did not cause A • A=>B must be equal to ~(B=>A)

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