 To combine propositions using connectives  To construct the truth table of a given compound proposition  To define de Morgan Law for logic  To define the difference between a predicate and a proposition  To use a quantifier in a predicate 1 RMIT University; Taylor's College LECTURE 7

 Let P be a proposition.  It has a truth value of T (for true) or F (for false).  Its negation is “not P”, denoted by ~P. RMIT University; Taylor's College 2 NEGATION: A REVISION P~P~P TF FT When P is true, ~P is false. When P is false, ~P is true. Truth table for Negation

Let P and Q be propositions. They can be combined in various ways.  Conjunction (AND)  Disjunction (OR)  Implication (if P then Q)  Equivalence (P if and only if Q) RMIT University; Taylor's College 3 COMBINING PROPOSITIONS

 Expressions created using the five connectives are called compound propositions.  We combine elementary (or constituent) propositions to create compound propositions.  The truth values of the constituent propositions determine the truth values of a compound proposition. RMIT University; Taylor's College 4 COMPOUND PROPOSITIONS

RMIT University; Taylor's College 5 TRUTH TABLES PQP Q TTFFTTFF TFTFTFTF TFFFTFFF PQ TTFFTTFF TFTFTFTF PQ TTFFTTFF TFTFTFTF PQ TTFFTTFF TFTFTFTF

a)Construct the truth table for the compound proposition b)Construct the truth table for the compound proposition RMIT University; Taylor's College 6 EXAMPLE 1

a)Construct the truth table for the compound propositions and. b)What is the relationship between these two compound propositions? RMIT University; Taylor's College 7 EXAMPLE 2

 Consequence: Any expression using “or” can be replaced by an expression using “and” and “not”  “I’ll watch G. I. Joe or Final Destination 4 this weekend” is logically equivalent to  “It’s not true that I won’t watch G.I. Joe and that I won’t watch Final Destination 4 this weekend” RMIT University; Taylor's College 8 DE MORGAN LAWS FOR LOGIC

 Example 3: Construct the truth table for the following compound proposition. RMIT University; Taylor's College 9 HIERARCHY OF CONNECTIVES

 If the last column in a truth table has only T (for true), then the compound proposition is called a tautology  If the last column in a truth table has only F (for false), then the compound proposition is called a contradiction  If a compound proposition is neither a tautology nor a contradiction then the last column of the truth table will have both T and F appearing. Such a compound proposition is called a contingency RMIT University; Taylor's College 10 TAUTOLOGY, CONTRADICTION, CONTINGENCY

Show that the following is a contradiction. RMIT University; Taylor's College 11 EXAMPLE 4

Several logical equivalences have been established RMIT University; Taylor's College 12 LOGICAL EQUIVALENCES

 Consequence: All compound propositions can be expressed using only two connectives: negation and conjunction, or negation and disjunction, or negation and implication  Example: “If dogs have humps then the moon is green” is equivalent to “Dogs don’t have humps or the moon is green”. RMIT University; Taylor's College 13 LOGICAL EQUIVALENCES

 A proposition has to be unambiguously true or false.  In contrast, a predicate is a statement involving at least one variable, for example, the variable x. The truth value may depend on the value of x.  Example: Let P(x) mean “x is an integer”. Then P(2) is true, but P(π) is false.  Example: Let Y(t) mean “ my friend t wears glasses ”. Then Y(Albert) may be true, while Y(Aaron) may be false. RMIT University; Taylor's College 14 PROPOSITIONS VS. PREDICATES

 What values is the variable allowed to take?  There may be several variables. Example, let T(x, y) mean x and y are relatively prime.  The variables have to range over some set D, called the domain of interpretation or the universe of discourse. RMIT University; Taylor's College 15 PREDICATE LOGIC

Let the domain of interpretation for a predicate T(x, y) be D = Z = the set of all integers.  Let T(x, y) mean that x and y are relatively prime.  Then T(10, 21) is true while T(12, 15) is false.  Why? RMIT University; Taylor's College 16 EXAMPLE 5

 It’s natural to introduce the idea of a quantifier when we’re considering predicates. These tell us how often the predicate is true  “for all”  “there exists”  Predicate logic involves statements like this:  [ for all x, P(x) is true ]  [ for all x there exists y such that P(x, y) is true ] RMIT University; Taylor's College 17 QUANTIFIERS