1.  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

2.  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

3. 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

4.  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

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

6. 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

7. 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

8.  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

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

10.  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

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

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

13.  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

14.  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

15.  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

16. 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

17.  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