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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
08 March 2009 Instructor: Tasneem Darwish 2
Outlines
Logical Equivalence.Logical implications.The algebra of Propositions.More about conditionals.Arguments.
08 March 2009 Instructor: Tasneem Darwish 3
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
08 March 2009 Instructor: Tasneem Darwish 4
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
08 March 2009 Instructor: Tasneem Darwish 5
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
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Example 1.5
We need to prove that the following two propositions are logically equivalent: p →(q →r ) (p ∧ q) → r
Logical Equivalence
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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
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If we have ‘P ├ Q’ then ‘P → Q’ is a tautology and vice versa.
Logical Implication
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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
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The following is a list of some important logical equivalences:
Idempotent laws Associative laws
Commutative laws Absorption laws
The Algebra of Propositions
08 March 2009 Instructor: Tasneem Darwish 11
The following is a list of some important logical equivalences:
Distributive laws involution law
De Morgan's laws Identity laws
Complement laws
The Algebra of Propositions
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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
The Algebra of Propositions
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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.
The Algebra of Propositions
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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
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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
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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
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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
08 March 2009 Instructor: Tasneem Darwish 19
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
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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
08 March 2009 Instructor: Tasneem Darwish 21
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