Chapter 1 The Logic of Compound Statements Section

Chapter 1 The Logic of Compound Statements

Section 1. 1 Logical Form and Logical Equivalence

Statements • A statement is a sentence that is either true or false, but not both. • Statements: – It is raining. – I am carrying an umbrella. • Not statements – He has a driver’s license. – Are you there? –x+y>0

Logical Operators • Binary operators – Conjunction – “and”. – Disjunction – “or”. • Unary operator – Negation – “not”. • Other operators – XOR – “one or the other but not both”” – NAND – “not both” – NOR – “neither”

Logical Symbols • Statements are represented by letters: p, q, r, etc. • means “and”. • means “or”. • means “not”.

Examples • Basic statements – p = “It is raining. ” – q = “I am carrying an umbrella. ” • Compound statements – p q = “It is raining and I am carrying an umbrella. ” – p q = “It is raining or I am carrying an umbrella. ” – p = “It is not raining. ”

Examples • But compound statements – “it is not hot but it is sunny. ” – but in this case is ^ “and” – p = “it is not hot” – q = “it is sunny” – expression: p ^ q – “it is not hot and it is not sunny”

Truth Table of an Expression • Make a column for every variable. • List every possible combination of truth values of the variables. • Make one more column for the expression. • Write the truth value of the expression for each combination of truth values of the variables.

Truth Table for “AND” • p q is true if p is true and q is true. • p q is false if p is false or q is false. p q T T F F F T F F

Truth Table for “OR” • p q is true if p is true or q is true. • p q is false if p is false and q is false. p q T T F F F

Truth Table for “not” • p is true if p is false. • p is false if p is true. p p T F F T

Example: Truth Table • Truth table for the statement ( p) (q r ). p q r ( p) (q r ) T T T F F T F T F F T T F F F T

Logical Equivalence • Two statements are logically equivalent if they have the same truth values for all combinations of truth values of their variables.

Example: Logical Equivalence • (p q) ( p q) p q (p q) ( p q) T T T F F F F T T

De. Morgan’s Laws • (p q) ( p) ( q) • It is not true that “John is short and he is fat”, then it is true that “John is not short or John is not fat”. • If it is not true that x 5 or x 10, then it is true that x > 5 and x < 10.

Tautologies and Contradictions • A tautology is a statement that is logically equivalent to T. • A contradiction is a statement that is logically equivalent to F. • Some tautologies: – p p – p q ( p q) • Some contradictions: – p p – p q ( p q)

Wrapup • Quiz on Tuesday (Chapter 1) • Homework due Thursday