Boolean Algebra. Topics Binary Number System AND, OR, NOT, Exclusive OR, and Implications operations Truth Tables and Algebraic Laws Boolean Logic and.

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 BINARYDECIMAL 00 11 102 113 1004 1015 1106 1117 10008 10019 101010 101111 110012 110113 111014 111115 1000016

AND FIRST NUMBERSECOND NUMBERRESULT 000 010 100 111 Usually denoted with ^ Example: 1 ^ 1 = 1 1 ^ 0 = 0

OR FIRST NUMBERSECOND NUMBERRESULT 000 011 101 111 Usually denoted with Example: 1 1 = 1 0 0 = 0

Example 1 For x = 1 and y = 0 (x ^ y ) (x ^ y) (1 ^ 0) (1 ^ 0) 0 0 0

NOT NUMBERResult 10 01 Usually denoted with ~ Example: ~1 = 0 ~0 = 1

EXCLUSIVE OR FIRST NUMBERSECOND NUMBERRESULT 000 011 101 110 Usually denoted with Example: 1 1 = 0 0 1 = 1 c

Example 2 For x = 1, y = 1, z = 0 ~(x ^ z) ( y ~z) ~(1 ^ 0) ( 1 ~0) ~0 ( 1 1) 1 1 0

Implication FIRST NUMBERSECOND NUMBERRESULT 001 011 100 111 Usually denoted with => Example: 1 => 1 = 1 0 => 1 = 1

Truth Tables Table that lists ALL possible values for a particular equation. Ex: (x ^ y ) (x y) xy(x^y) (x y)(x^y) (x y) 00000 01010 10010 11111

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)

Application 1: Circuits

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

Example: Alarm

Example: Half Adder Adds 2 bits 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 0 + 1 (carry_ 0 + 0 + 1 = 1

Application 2: Searching

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.

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.

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

Example 2: Searching Techniques Lets search for graduate school that are NOT in Alabama, but are in Georgia OR Florida graduate schools -"Alabama" +"Florida" OR +"Georgia"

Application 3: Analyzing Fallacies

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. Lets test that out! ABA=>BB=>A 0011 0110 1001 1111

Fallacy II Is this a fallacy? A cause B therefore B did not cause A A=>B must be equal to ~(B=>A) ABA=>BB=>A~(B=>A) 00110 01101 10010 11110

Boolean Algebra. Topics Binary Number System AND, OR, NOT, Exclusive OR, and Implications operations Truth Tables and Algebraic Laws Boolean Logic and.

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