Download presentation

Presentation is loading. Please wait.

Published byRuby Casey Modified over 5 years ago

1
CSCI 115 Chapter 2 Logic

2
CSCI 115 §2.1 Propositions and Logical Operations

3
§2.1 – Propositions and Log Ops Logical Statement Logical Connectives –Propositional variables –Conjunction (and: ) –Disjunction (or: ) –Negation (not: ~) Truth tables

4
§2.1 – Propositions and Log Ops Quantifiers –Consider A = {x| P(x)} –t A if and only if P(t) is true –P(x) – predicate or propositional function Programming –if, while –Guards

5
§2.1 – Propositions and Log Ops Universal Quantification – true for all values of x – x P(x) Existential Quantification – true for at least one value – x P(x) Negation of quantification

6
CSCI 115 §2.2 Conditional Statements

7
§2.2 – Conditional Statements Conditional statement: If p then q –p q –p – antecedent (hypothesis) –q – consequent (conclusion) Truth tables

8
§2.2 – Conditional Statements Given a conditional statement p q –Converse –Inverse –Contrapositive Biconditional (if and only if) –p q is equivalent to ((p q) (q p))

9
§2.2 – Conditional Statements Statements –Tautology (always true) –Absurdity (always false) –Contingency (truth value depends on the values of the propositional variables) Logical equivalence ( )

10
CSCI 115 §2.3 Methods of Proof

11
§2.3 – Methods of Proof Prove a statement –Choose a method Disprove a statement –Find a counterexample Prove or disprove a statement –Where do I start?

12
§2.3 – Methods of Proof Direct Proof Proof by contradiction Mathematical Induction (§2.4)

13
§2.3 – Methods of Proof Valid rules of inference –((p q) (q r)) (p r) –((p q) p) q Modus Ponens –((p q) ~q) ~pModus Tollens –~~p pNegation –p ~~pNegation –p pRepitition Common mistakes – the following are NOT VALID –((p q) q) p –((p q) ~p) ~q

14
CSCI 115 §2.4 Mathematical Induction

15
§2.4 – Mathematical Induction If we want to show P(n) is true n Z, n > n 0 where n 0 is a fixed integer, we can do this by: i) Show P(n 0 ) is true Basic step ii) Show that for k > n 0, if P(k) is true, then P(k + 1) is true Inductive step

Similar presentations

© 2020 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google