MAT 3100 Introduction to Proof Logic Truth Tables

MAT 3100 Introduction to Proof Logic & Truth Tables

Logic = the study of correct reasoning p Use of logic p n In mathematics: p n to prove theorems In computer science: p to prove that programs do what they are supposed to do

Propositions p

Operations on Truth Values 1. Negation (Not), ~p, ¬p, p p Examples: p: I am 21. ~p: I am not 21. Rules n n If p is T, then ~p is __. If p is F, then ~p is __. Truth Table p T F ~p

Operations on Truth Values 2. Conjunction (And), p q Examples: p: I am going to town. q: It is going to rain. p q: I am going to town and it is going to rain. p Rules p q is T if both p and q are __. p

Operations on Truth Values 2. Conjunction (And), p q p q T T T F F p q

Operations on Truth Values 3. (Inclusive) Disjunction (Or), p q Examples: p: I am going to town. q: It is going to rain. p q: I am going to town or it is going to rain. p Rules p q is T if p or q is __. p

Operations on Truth Values 3. (Inclusive) Disjunction (Or), p q p q T T T F F p q

Operations on Truth Values 4. Exclusive Disjunction (Exclusive Or), p exor q p Rules p exor q is T if p or q is __, but not both.

Operations on Truth Values 5. Implication (If…then…), p q Examples: p: I am going to town. (hypothesis) q: It is going to rain. (conclusion) p q: If I am going to town then it is going to rain. p Rules p q is F only when p is __ but q is __. p

Operations on Truth Values 5. Implication (If…then…), p q Remark: Implication is defined different from normal (English) Language. p It is independent of __________ between the hypothesis and conclusion. p

Operations on Truth Values 5. Implication (If…then…), p q p q T T T F F p q

Operations on Truth Values 5. Implication (If…then…), p q If the hypothesis is F, then the implication is always T. p Examples: p: I have $1 million in the bank. q: I donate $10, 000 to my church. p q: p

Operations on Truth Values Given p q, we define the following related implications. a. Converse of p q : _______ b. Contrapositive of p q: _______ c. Inverse of p q: _______ We will see these related implications very often.