CSCI 115

Chapter 2

Logic

CSCI 115

§2.1

Propositions and Logical Operations

§2.1 – Propositions and Log Ops

• Logical Statement• Logical Connectives

– Propositional variables– Conjunction (and: )– Disjunction (or: )– Negation (not: ~)

• Truth tables

§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

§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

CSCI 115

§2.2

Conditional Statements

§2.2 – Conditional Statements

• Conditional statement: If p then q – p q– p – antecedent (hypothesis)– q – consequent (conclusion)

• Truth tables

§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))

§2.2 – Conditional Statements

• Statements– Tautology (always true)– Absurdity (always false)– Contingency (truth value depends on the values

of the propositional variables)

• Logical equivalence ()

CSCI 115

§2.3

Methods of Proof

§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?

§2.3 – Methods of Proof

• Direct Proof• Proof by contradiction• Mathematical Induction (§2.4)

§2.3 – Methods of Proof

• Valid rules of inference– ((p q) (q r)) (p r)– ((p q) p) q Modus Ponens– ((p q) ~q) ~p Modus Tollens– ~~p p Negation– p ~~p Negation– p p Repitition

• Common mistakes – the following are NOT VALID– ((p q) q) p– ((p q) ~p) ~q

CSCI 115

§2.4

Mathematical Induction

§2.4 – Mathematical Induction

• If we want to show P(n) is true nZ, n > n0 where n0 is a fixed integer, we can do this by:

i) Show P(n0) is true• Basic step

ii) Show that for k > n0, if P(k) is true, then P(k + 1) is true• Inductive step