Chapter 2.

Sanguk Noh

Logic

Propositions and logical operations

Conditional statements

Methods of proof

Mathematical Induction

Propositions and logical operations

Propositions and logical operations

Def.) statement or proposition

A declarative sentence that is either true or false, but not both.

e.g.)

statement Not a statement

The earth is round. Do you play tennis?

2+3=5 3-x=5

Take two aspirins.

Propositions and logical operations Logical connectives and compound statements

Propositional variables : can be replaced by statements

Compound statements : by logical connectives

syntax

Syntax

Negation ~p

Conjunction p∧q

Disjunction p∨q

Implication p⇒q

Equivalence p⇔q

Universal ∀x

Existential ∃x

Relation R(x,y)

Propositions and logical operations

Negation of p

Conjunction of p and q (⇒ and)

p ~p

T F

F T

~: not ⇒ a unary operation

p q p∧q

T T T

T F F

F T F

F F F

Propositions and logical operations

Disjunction of p and q (⇒ or)

e.g.) p : It is snowing. q: 3<5

p∧q : It is snowing and 3<5.

p∨q : It is snowing or 3<5.

p q p∨q

T T T

T F T

F T T

F F F

Propositions and logical operations

Quantifiers A predicate or a propositional function P(x)

{x | P(x)} - properties of objects or

- relations between objects

Universal quantification For all values of x, P(x) is true.

⇒ ∀x P(x)

Existential quantification There exists a value of x for which P(x) is true.

⇒ ∃x P(x)

Universal quantifier

existential quantifier

*negation ~(∀x P(x)) = ∃x~P(x) ~(∃x P(x)) = ∀x~P(x)

Conditional statements Conditional statements

Conditional statement or implication If p then q : p ⇒ q conclusion

hypothesis (premise)

p q p⇒q ~p∨q ~q⇒~p

T T T T T

T F F F F

F T T T T

F F T T T

A false hypothesis implies any conclusion.

~q⇒~p ≡ q∨~p ≡ ~p∨q Contrapositive of p⇒q

Conditional statements Equivalence (or biconditional)

p if and only if q : p⇔q

e.g.) 3>2 if and only if 0<(3-2)

p q p⇒q q⇒p p⇔q

T T T T T

T F F T F

F T T F F

F F T T T

Conditional statements Tautology

A statement that is “true for all possible values” of its propositional variables

Contradiction : always false

Contingency : either true or false e.g.)

(p⇒q)⇔(~p∨q) : tautology

p∧~p: contradiction (nil)⇒empty clause

(p⇒q)∧(p∨q) : contingency

p and q are logically equivalent, p≡q, if p⇔q is a tautology.

e.g.) p∨q≡q∨p, if (p∨q) ⇔ (q∨p) is a tautology.

Conditional statements Theorem 1. the operations for propositions

(a) commutativity p∨q ≡ q∨p, p∧q ≡ q∧p

(b) associativity p∨(q∨r)≡(p∨q)∨r, p∧(q∧r)≡(p∧q)∧r

(c) distributivity p∨(q∧r)≡(p∨q)∧(p∨r) p∧(q∨r)≡(p∧q)∨(p∧r)

(d) Idempotency p∨p≡p, p∧p≡p

(e) negation ~(~p)≡p, ~(p∨q)≡~p∧~q ~(p∧q)≡~p∨~q

Conditional statements Theorem 2. The implication operation

(a) (p⇒q) ≡ (~p∨q) ≡ (~q⇒~p)

(b) (p⇔q) ≡ (p⇒q)∧(q⇒p)

(c) ~(p⇒q) ≡ ~(~p∨q) ≡ p∧~q

(d) ~(p⇔q) ≡ ~(~p∨q)∨~(~q∨p)

≡ (p∧~q)∨(q∧~p)

Conditional statements Formal system or formal languages

Propositional logic : facts(exists in the world)

First-order predicate logic : facts, objects, relations

Objects(things) : people, numbers, colors…..

Relations : brother of, owns, bigger than, equals..

Functions : plus, cosine, father of,….

We can quantify over objects but not over relations or functions on those objects.

Conditional statements

Theorem 3. The universal and existential quantifiers (a) ~(∀xP(x)) ≡ ∃x~P(x)

(b) ~(∃xP(x)) ≡ ∀x~P(x)

(c) ∀xP(x) ≡ ~∃x~P(x) e.g.) ∀x Likes(x, IceCream) ≡ ~∃x ~Likes(x, IceCream)

Everyone likes ice cream.

There is no one who does not like ice cream.

(d) ∃xP(x) ≡ ~∀x~P(x) e.g.) ∃x Likes(x, IceCream) ≡ ~∀x ~Likes(x, IceCream)

Someone likes ice cream.

Not everyone doesn’t like ice cream.

Methods of proof Methods of proof

q logically follows from p, if p⇒q is a tautology.

e.g.) (p1∧ p2∧… ∧ pn)⇒q : tautology

hypotheses P1

P2

⋮ pn

∴ q

conclusion

q logically follows from p 1, p2, …, pn

*mathematical proofs: if the p i‘s are all true, q has to be true. As beginning w/ the hypotheses, →each step is justified by rules of inference, →arriving at the conclusion.

Methods of proof Theorem 4. inference rules : tautology

(a) (p∧q)⇒p (b)(p∧q)⇒q

→ and-elimination

~(p∧q)∨p=~p∨~q∨p

~(p∧q)∨q=~p∨~q∨q

(c) p⇒(p∨q) (d) q⇒(p∨q)

→ or-introduction

~p∨p∨q ~q∨p∨q

Methods of proof (e) ~p⇒(p⇒q) (f) ~(p⇒q)⇒p

p∨(~p∨q) ~p∨q∨p

(g) (p∧(p⇒q)) ⇒ q

→modus ponens

=~(p∧(~p∨q))∨q

=(~p∨(p∧~q))∨q

=((~p∨p)∧(~p∨~q))∨q

=(~p∨~q)∨q=~p∨(~q∨q)

p p→q ∴q

Methods of proof (h)(~p∧(p∨q))⇒q = ~(~p∧(p∨q))∨q

= p∨~(p∨q)∨q

= (p∨q)∨~(p∨q)

(i) (~q∧(p⇒q)) ⇒~p

=~(~q∧(~p∨q))∨~p

=q∨~(~p∨q)∨~p=(q∨~p)∨~(q∨~p)

(j) ((p⇒q)∧(q⇒r)) ⇒ (p⇒r)

((~p∨q)∧(~q∨r)) ⇒ ~p∨r

→Unit resolution

→resolution

Mathematical induction We wish to show that P(n) is true ∀n≥n0, where n0

is some fixed integer.

(1) 『In the basis step we show that P(n0)is true.』

That is, P is true of n0.

(2) 『 The induction hypothesis is the assumption that for

some fixed but arbitrary k≥n0, P(k) is true. 』

(3) 『 In the induction step we show, using the induction

hypothesis, that P(k+1) must also be true. 』