MAT 3100 Introduction to Proof

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Presentation transcript:

MAT 3100 Introduction to Proof Logic & Truth Tables

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

Propositions A proposition is a statement or sentence that can be determined to be either true (T) or false (F). Examples: 1+1=2 3+1=2 My dog ate my homework. 𝑛+1>3 4𝑥2+2𝑦−1

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

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. Rules p  q is T if both p and q are __.

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

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. Rules p  q is T if p or q is __.

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

Operations on Truth Values 4. Exclusive Disjunction (Exclusive Or), p exor q 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. Rules p  q is F only when p is __ but q is __.

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

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

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

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.

Operations on Truth Values Examples: If 𝑥=𝑦 then 𝑥2=𝑦2 Converse: If 𝑥2=𝑦2 then 𝑥=𝑦 Contrapositive: If 𝑥2≠𝑦2 then 𝑥≠𝑦 Inverse: If 𝑥≠𝑦 then 𝑥2≠𝑦2

Operations on Truth Values Examples: If 𝑥=5 then 𝑥+1=6 Contrapositive: If 𝑥+1≠6 then 𝑥≠5

Operations on Truth Values Examples: If 𝑥=1 then 𝑥2=1 Inverse: If 𝑥≠1 then 𝑥2≠1

Operations on Truth Values 6. Double Implication, Biconditional Proposition, (…if and only if…), p  q Rules p  q is T if p and q ___________________.

Operations on Truth Values Summary p q p  q p  q p  q p  q T F

In Class Learning Activity Classwork Group Explorations