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

08 March 2009 Instructor: Tasneem Darwish 1

University of PalestineFaculty of Applied Engineering and Urban Planning

Software Engineering Department

Introduction to Discrete Mathematics

Propositional LogicPart 2


Outlines

Logical Equivalence.Logical implications.The algebra of Propositions.More about conditionals.Arguments.


Two propositions are said to be logically equivalent if they have identical truth values for every set of truth values of their components.

Using P and Q to denote (possibly) compound propositions, we write P ≡ Q if P and Q are logically equivalent.

Example 1.4 Show that

Logical Equivalence


if two compound propositions are logically equivalence (P ≡ Q) then P ↔ Q is a tautology. Because two logically equivalent propositions are either both true or both false.

if P ↔ Q is a tautology, then P ≡ Q.

Logical Equivalence


Example 1.5Show that the following two propositions are logically equivalent.(i) If it rains tomorrow then, if I get paid, I’ll go to Paris.(ii) If it rains tomorrow and I get paid then I’ll go to Paris.

Solution: Define the following simple propositions:

p : It rains tomorrow. q : I get paid. r : I’ll go to Paris.

The first sentence can be written as p →(q →r ) The second sentence can be written as (p ∧ q) → r

Logical Equivalence


Example 1.5

We need to prove that the following two propositions are logically equivalent: p →(q →r ) (p ∧ q) → r

Logical Equivalence


A proposition P is said to logically imply a proposition Q if, whenever P is true, then Q is also true

‘P logically implies Q’ is written as P ├ Q.

Example 1.6 show that

whenever q is true (second and third rows), p ∨ q is also true.

Logical Implication


If we have ‘P ├ Q’ then ‘P → Q’ is a tautology and vice versa.

Logical Implication


Example 1.7 Show that (p ↔ q) ∧ q logically implies p.

we can show that [(p ↔ q) ∧ q] ├ p in one of two ways:We can show that p is always true when (p ↔ q) ∧ q is true we can show that [(p ↔ q) ∧ q] → p is a tautology.

The truth table for (p ↔ q) ∧ q is given by:

Logical Implication


The following is a list of some important logical equivalences:

Idempotent laws Associative laws

Commutative laws Absorption laws

The Algebra of Propositions


The following is a list of some important logical equivalences:

Distributive laws involution law

De Morgan's laws Identity laws

Complement laws



The Duality PrincipleGiven any compound proposition P involving only the connectives denoted by ∧ and ∨, the dual of that proposition is obtained by: replacing ∧ by ∨ replacing ∨ by ∧ replacing t by f replacing f by t.

Example: The dual of (p ∧ q)∨ ￣ p is (p ∨ q)∧ ￣ p.The dual of (p ∨ f ) ∧ q is (p ∧ t) ∨ q.

•The duality principle states that, if two propositions are logically equivalent, then so are their duals



Replacement RuleSuppose that we have two logically equivalent propositions P1

and P2, so that P1 ≡ P2.Suppose also that we have a compound proposition Q in which

P1 appears.

The replacement rule says that we may replace P1 by P2 and the resulting proposition is logically equivalent to Q.



Example 1.8Prove that



Given the conditional proposition p →q, we define the following:

the converse of p → q is: q → pthe inverse of p →q is: ￣ p → ￣ qthe contrapositive of p →q is: ￣ q → ￣ p

Note:a conditional proposition p →q and its contrapositive ￣ q

→ ￣ p are logically equivalent (i.e. (p → q) ≡( ￣ q → ￣p)).

More about Conditionals


Example 1.9State the converse, inverse and contrapositive of the proposition ‘If Jack plays his guitar then Sara will sing’.Solution: We define:

p: Jack plays his guitarq: Sara will sing

p →q: If Jack plays his guitar then Sara will sing.

Converse: q → p: If Sara will sing then Jack plays his guitar.Inverse: ￣ p → ￣ q: If Jack doesn’t play his guitar then Sara won’t sing.Contrapositive: ￣ q → ￣ p: If Sara won’t sing then Jack doesn’t play his guitar.

More about Conditionals


An argument consists of:a set of propositions called premisesanother proposition, supposed to result from the premises,

called the conclusion.

Thus if we have premises P1, P2, . . . , Pn and a conclusion Q.We say that the argument is valid if:

(P1 ∧ P2 ∧・・・∧ Pn) ├ Q, or(P1 ∧ P2 ∧・・・∧ Pn) → Q is a tautology.

Thus, whenever P1, P2, . . . , Pn are all true, then Q must be true.

Arguments


Examples 1.101) Test the validity of the following argument: ‘If you insulted Bob then I’ll never speak to you again. You insulted Bob so I’ll never speak to you again.’

Solution: We define: p: You insulted Bob.q: I’ll never speak to you again.

The premises in this argument are: p →q and p.The conclusion is: q.We must investigate the truth table for [(p → q) ∧ p] →q to see

whether it is a tautology or not.

Arguments


Examples 1.101) Test the validity of the following argument: ‘If you insulted Bob then I’ll never speak to you again. You insulted Bob so I’ll never speak to you again.’

SolutionWe must therefore investigate the truth table for [(p → q) ∧ p] →q to see whether it is a tautology or not.

Arguments


Examples 1.102)Test the validity of the following argument:‘If you are a mathematician then you are clever. You are clever

and rich. Therefore if you are rich then you are a mathematician.’

Solution Define: p: You are a mathematician.q: You are clever.r : You are rich.

The premises are: p →q and q ∧ r .The conclusion is: r → p.We must test whether or not [(p →q) ∧ (q ∧ r )] → (r → p) is a

tautology.

Arguments


Examples 1.102)Test the validity of the following argument:‘If you are a mathematician then you are clever. You are clever

and rich. Therefore if you are rich then you are a mathematician.’

Solution We must test whether or not [(p →q) ∧ (q ∧ r )] → (r → p) is a

tautology.

Arguments

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

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