CS1231 Tutorial 2

CS1231 Discrete Structures

Tong W. Ratanapan

Tutorial 2: Sep 2-6, 2013

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http://www.slideshare.net/TongWR

Outline Tutorial 1 Revision

Discussion

CS1231 Tutorial 2 Tong W. Ratanapan

Correction

Consultation

Launchpad @ N-House (level 1, block 15, PGP)

Wednesday 1-2pm


Discussion


Outline


Discussion


0. Outline

1. Tutorial 1 1. Pigeon Hole Principle 2. Pure Existence Proof 3. Inclusion-Exclusion Principle


Discussion


0. Outline


2. Revision 1. Propositional Logic 2. Predicate Logic 3. Model Semantics


Discussion


0. Outline


2. Revision 1. Propositional Logic 2. Predicate Logic 3. Model Semantics

3. Discussion


Discussion

Pigeonhole Principle Pure Existence Proof Inclusion-Exclusion Principle


Tutorial 1


Discussion




Discussion



Problems & Solutions: http://www.artofproblemsolving.com/Wiki/index.php/Pigeonhole_Principle Applications: http://mindyourdecisions.com/blog/2008/11/25/16-fun-applications-of-the-pigeonhole-principle

http://www.artofproblemsolving.com/Wiki/index.php/Pigeonhole_Principle

http://www.artofproblemsolving.com/Wiki/index.php/Pigeonhole_Principle

http://mindyourdecisions.com/blog/2008/11/25/16-fun-applications-of-the-pigeonhole-principle














Discussion



Pure Existence Proof


Discussion




Q: Are there irrational 𝑝, 𝑞 such that 𝑝𝑞 is rational?


Discussion




Q: Are there irrational 𝑝, 𝑞 such that 𝑝𝑞 is rational? A: Yes!


Discussion





22


Discussion





22

rational


Discussion





22

rational 𝑝 = 𝑞 = 2 → 𝑝𝑞 = 22


Discussion





22


; rational by assumption


Discussion





22

irrational




Discussion





22

irrational



𝑝 = 22, 𝑞 = 2 → 𝑝𝑞 = 2

22

= 2 ; rational


Discussion





22

irrational



𝑝 = 22, 𝑞 = 2 → 𝑝𝑞 = 2

22

= 2 ; rational

doesn’t matter, exists anyway!


Discussion



Inclusion-Exclusion Principle


Discussion

Propositional Logic Predicate Logic Model Semantics


Revision


Discussion



Propositional Logic

Propositional Logic Formal Syntax of Propositional Logic Predicate Logic Formal Syntax of Predicate Logic

Formal Syntax of Propositional Logic

2.2. Formal Syntax of Propositional Logic

Definition 2.2.1

Let us consider a collection of symbols called propositions orstatements. Let us consider the collection of symbols{>,⊥,∨,∧,¬,⇒,⇔,⊕}a. Let us consider the two parenthesissymbols ( and ).

F is a well formed formula (wff) (or atomic statementb) if F isa proposition.

The following are wff (or compound statements) iff F1 and F2are wff. >, ⊥, (F1 ∧ F2), (F1 ∨ F2), ¬(F1), (F1 ⇒ F2),(F1 ⇔ F2), (F1 ⊕ F2).

the above are the only wff.

aThese are the standard connectives. There are more.bp and ¬p are sometimes refer to as unit statements.


Formal Syntax of Propositional Logic

Different Notations

> T , true, 1⊥ F , false, 0

(p ∧ q) (p.q), p × q, (p&q), (p&&q)(p ∨ q) (p + q), (p | q), (p‖q)(p ⊕ q) (pˆq)(p ⇔ q) (p ↔ q), (p ≡ q)(p ⇒ q) (p → q), (p ⊃ q)¬p ∼ p, p, !p(p) [p]


Precedence

Precedence and Parenthesis

In this module we do not assume any precedence rule andsystematically use parenthesis to avoid ambiguity.

Truth Tables Propositional Calculus Predicate Calculus

Truth Tables

Truth Table of Negation

F1 ¬F1true falsefalse true


Truth Tables

Truth Table of Conjunction

F1 F2 F1 ∧ F2

true true truetrue false falsefalse true falsefalse false false


Truth Tables

Truth Table of Disjunction

F1 F2 F1 ∨ F2

true true truetrue false truefalse true truefalse false false


Truth Tables

Truth Table of Exclusive Disjunction

F1 F2 F1 ⊕ F2

true true falsetrue false truefalse true truefalse false false


Truth Tables

Truth Table of Implication

F1 F2 F1 ⇒ F2

true true truetrue false falsefalse true truefalse false true


Truth Tables

Truth Table of Equivalence

F1 F2 F1 ⇔ F2

true true truetrue false falsefalse true falsefalse false true


Discussion



Predicate Logic


Formal Syntax of Predicate Logic

Definition 2.4.5

Let us consider a collection of constants. Let us consider acollection of variables, let us consider a collection of predicates(each predicate is associated to a natural number n, called itsvalence or arity, we say that the predicate is n-ary). Let us considerthe corresponding Herbrand’s base. Let us consider the collectionof symbols {>,⊥,∨,∧,¬,⇒,⇔,⊕}. Let us consider the twosymbols ∃ (the existential quantifier; read “there exists”) and ∀(the universal quantifier; read “for all”) called quantifiers....



Definition 2.4.5 Cont.

F is a well formed formula (wff) or compound statement if Fis an atom of BH .

The following are wffs if F1 and F2 are wffs.

>,⊥,(F1 ∧ F2),(F1 ∨ F2),¬(F1),(F1 ⇒ F2),(F1 ⇔ F2),(F1 ⊕ F2).

The following are wffs if F is a wff and X is a variable.

(∀X F ),(∃X F ).

the above are the only wff.



For example the following are wff.

∀X (∀Y ((odd(X ) ∧ ¬(even(Y )))⇒ even(X + Y ))).

∀X (∃Y (X ∈ N⇒ (Y ∈ N ∧ Y = s(X )))).

∀X (X ∈ N⇒ s(X ) ∈ N).

For example the following are not wff.

∀X (∀Y ((odd(X ) ∧ ∧ ∧ ¬(even(Y )))⇒ even(X + Y ))).

∀X ∧ (∃Y (X ∈ N⇒ (Y ∈ N ∧ Y = s(X )))).

X (X ∈ N⇒ s(X ) ∈ N.


Discussion



Model Semantics


Propositional Calculus

3.2. Propositional Calculus

Definition 3.2.1

An interpretation I is a mapping of the propositions to the truthvalues {true, false} such that:

I (>) = true

I (⊥) = false,

I (¬F1) = true if I (F1) = false , otherwise I (¬F1) = false,

I (F1 ∧ F2) = true if I (F1) = true and I (F2) = true, otherwiseI (F1 ∧ F2) = false,

I (F1 ∨ F2) = true if I (F1) = true or I (F2) = true (or both),otherwise I (F1 ∨ F2) = false,

...


Propositional Calculus

Definition Cont.

I (F1 ⇒ F2) = false if I (F1) = true and I (F2) = false,otherwise I (F1 ⇒ F2) = true,

I (F1 ⇔ F2) = true if I (F1) = true = I (F2) = true orI (F1) = false = I (F2) = false, otherwise I (F1 ⇔ F2) = false,

I (F1 ⊕ F2) = true if I (F1) = true or I (F2) = true but notboth, otherwise I (F1 ⊕ F2) = false.

The above is best summarized in truth tables. A truth tablepresents all interpretations of a formula.


Models

Definition 3.2.2

An interpretation I is a model of a formula F iff I (F ) = true.

Definition 3.2.3

An interpretation I is a counter-model of a formula F iffI (F ) = false.


Models

Definition 3.2.4

A formula F is a tautology (or is valid) iff all interpretations aremodels.

Definition 3.2.5

A formula F is a contradiction iff no interpretation is model.

Definition 3.2.6

A formula F is a contingency iff it is neither a tautology nor acontradiction.


Models

Definition 3.2.7

Let F1 and F2 be two formulae. F2 is a logical consequence of F1,we write F1 |= F2 iff all models of F1 are models of F2.

Definition 3.2.8

Let Fn be formulae. Fn is a logical consequence of F1, ...,Fn−1, wewrite F1, ...,Fn−1 |= Fn iff all models common to F1, ...,Fn−1 aremodels of Fn.

Definition 3.2.9

Let F1 and F2 be two formulae F1 and F2 are logicaly equivalent,we write F1 ≡ F2, iff they have the same models.


Discussion

Group activity


Discussion


Discussion

Group activity


Divide into 5 groups

Each solves one from 2, 3, 4, 5.odd, 5.even

Education

CS1231 Tutorial 2