04 march 2009instructor: tasneem darwish1 university of palestine faculty of applied engineering and...

04 March 2009 Instructor: Tasneem Darwish 1

University of PalestineFaculty of Applied Engineering and Urban Planning

Software Engineering Department

Introduction to Discrete Mathematics

Propositional Logic


Outlines

Propositions and Truth Values.Logical Connectives and Truth Tables.Tautologies and Contradictions.


Logic is used to determine the validity of arguments.

Logic provides rules that can be used to judged an argument as sound or unsound.

A proposition is a declarative statement which is either true or false, but not both.

Examples of propositions: Triangles have four vertices. 3 + 2 = 4. 6 < 24. Tomorrow is Thursday.

Propositions and Truth Values


Exclamations, questions and demands are not propositions since they cannot be declared true or false.

The following are not propositions: Keep off the grass. That’s Fantastic! Did you go to Jane’s party? Don’t say that.

The truth (T) or falsity (F) of a proposition is called truth value.

Propositions and Truth ValuesPropositions and Truth ValuesPropositions and Truth ValuesPropositions and Truth Values


Propositions are conventionally symbolized using the letters p, q, r, . .

Example:

p: Manchester is in Scotland.

q: Apples are fruits.

b: Today is Wednesday.

Propositions and Truth Values


A simple proposition consist of a single statement.

A Compound proposition is a combination of simple propositions linked using logical connectives.

The truth value of any compound proposition is determined by:

The truth values of its component simple propositions.

The particular connective, or connectives, used to link them.

Logical Connectives and Truth Tables


All connectives are performed on more than one proposition except the negation operation.

The negation operation has the effect of reversing the truth value of the proposition.

The negation of a proposition p is written as ￣ p (or ~p or ￢ p)

The truth table for the negation operation:



The logical connectives that can be used to link pairs of propositions are:

Conjunction

Inclusive Disjunction.

Exclusive Disjunction.

The conditional connective.

The bi-conditional connective.



Any two simple propositions can be combined by using the word ‘and’ between them.

The resulting compound proposition is called the conjunction of its two simple propositions component.

Example :p : The sun is shining.q : children eat ice cream.p q : The sun is shining and children eat ice cream.∧Or p . q: The sun is shining and children eat ice cream.

Conjunction


The following truth table gives the truth values of p q.∧

the conjunction p q is true only when both p and q are ∧true.

Conjunction


Any two simple propositions can be combined by using the word ‘or’ between them.

The resulting compound proposition is called the Disjunction of its two simple propositions component.

There are two types of disjunction:Inclusive Disjunction.Exclusive Disjunction.

Disjunction


For propositions p and q, p q∨ symbolizes the inclusive disjunction of p and q.

The inclusive disjunction is true when either or both of its components are true and is false otherwise.

The truth table for p q is∨

Disjunction


For propositions p and q, symbolizes the exclusive disjunction of p and q.

The exclusive disjunction is true when exactly one of its components is true.

The truth table for is

Disjunction


The conditional connective (implication) is symbolized by→.

The linguistic expression of a conditional proposition is normally ‘if . . . then . . . ’ as in the following example:

p : I eat breakfast.q : I don’t eat lunch.p → q : If I eat breakfast then I don’t eat lunch.

Alternative expressions for p →q are:I eat breakfast only if I don’t eat lunch.Whenever I eat breakfast, I don’t eat lunch.That I eat breakfast implies that I don’t eat lunch.

The Conditional connective


The conditional connective truth table is as follows:

The proposition p is sometimes called the antecedent and q the consequent.

The proposition p is said to be a sufficient condition for q and q a necessary condition for p.

False is considered stronger than True.If the antecedent is stronger than or equal to the

consequent, then the conditional connective value is TRUE

The Conditional connective


The biconditional connective is symbolized by↔, and expressed by ‘if and only if . . . then . . . ’. Example:

p : I eat breakfast.q : I don’t eat lunch.p ↔ q : I eat breakfast if and only if I don’t eat lunch (or

alternatively, ‘If and only if I eat breakfast, then I don’t eat lunch’).

The truth table for p ↔ q is given by:for p ↔ q to be true, p and q must

both have the same truth values.

The Biconditional connective


Consider the following propositions:p : Mathematicians are generous. q : Spiders hate algebra.Write the compound propositions symbolized by:(i) p ∨ ￣ q (ii) (q p)∧(iii) ￣ p →q (iv) ￣ p ↔ ￣ q.

Solution(i) Mathematicians are generous or spiders don’t hate algebra (or both).(ii) It is not the case that spiders hate algebra and mathematicians are generous.(iii) If mathematicians are not generous then spiders hate algebra.(iv) Mathematicians are not generous if and only if spiders don’t hate algebra.

Examples 1.1


Consider the propositionsp: ‘Today is Monday’ q :‘I’ll go to London’.Write the symbols for the following propositions:(i) If today is Monday then I won’t go to London.(ii) Today is Monday or I’ll go to London, but not both.(iii) I’ll go to London and today is not Monday.(iv) If and only if today is not Monday then I’ll go to London.

Solution(i) p → ￣ q (ii)

(iii) q ∧ ￣ p (iv) ￣ p ↔ q.

Examples 1.1


Construct truth tables for the following compound propositions.

(i) ￣ p q∨ (ii) ￣ p ∧ ￣ q

(iii) ￣ q → p (iv) ￣ p ↔ ￣ q.

Examples 1.1


Construct truth tables for:(i) p →(q r )∧ (ii) ( ￣ p q) ↔ ∨ ￣ r .

Examples 1.1


A tautology is a compound proposition which is true no matter what the truth values of its simple components.

A contradiction is a compound proposition which is false no matter what the truth values of its simple components.

Example 1.2: Show that the following propositions are tautologies:

p ∨ ￣ p (p q) (p q)∧ ∨ ∧

Tautologies and Contradictions


Example 1.3: Show that (p ∧ ￣ q) ( ∧ ￣ p q) is a ∨contradiction

Tautologies and Contradictions

04 march 2009instructor: tasneem darwish1 university of palestine faculty of applied engineering and...

Documents

truth values of p

proposition p

truth tables

tasneem darwish5 propositions

conjunction slide

conjunction p q

pairs of propositions

examples of propositions