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08 March 2009Instructor: Tasneem Darwish1 University of Palestine Faculty of Applied Engineering and Urban Planning Software Engineering Department Introduction to Discrete Mathematics Propositional Logic Part 2

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08 March 2009Instructor: Tasneem Darwish2 Outlines Logical Equivalence. Logical implications. The algebra of Propositions. More about conditionals. Arguments.

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08 March 2009Instructor: Tasneem Darwish3 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

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08 March 2009Instructor: Tasneem Darwish4 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

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08 March 2009Instructor: Tasneem Darwish5 Example 1.5 Show 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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08 March 2009Instructor: Tasneem Darwish6 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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08 March 2009Instructor: Tasneem Darwish7 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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08 March 2009Instructor: Tasneem Darwish8 If we have ‘P ├ Q’ then ‘P → Q’ is a tautology and vice versa. Logical Implication

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08 March 2009Instructor: Tasneem Darwish9 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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08 March 2009Instructor: Tasneem Darwish10 The following is a list of some important logical equivalences: Idempotent lawsAssociative laws Commutative lawsAbsorption laws The Algebra of Propositions

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08 March 2009Instructor: Tasneem Darwish11 The following is a list of some important logical equivalences: Distributive lawsinvolution law De Morgan's lawsIdentity laws Complement laws The Algebra of Propositions

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08 March 2009Instructor: Tasneem Darwish12 The Duality Principle Given 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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08 March 2009Instructor: Tasneem Darwish13 Replacement Rule Suppose 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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08 March 2009Instructor: Tasneem Darwish14 Example 1.8 Prove that The Algebra of Propositions

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08 March 2009Instructor: Tasneem Darwish15 Given the conditional proposition p →q, we define the following: the converse of p → q is:q → p the inverse of p →q is: ￣ p → ￣ q the 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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08 March 2009Instructor: Tasneem Darwish16 Example 1.9 State the converse, inverse and contrapositive of the proposition ‘If Jack plays his guitar then Sara will sing’. Solution: We define: p: Jack plays his guitar q: 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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08 March 2009Instructor: Tasneem Darwish17 An argument consists of: a set of propositions called premises another 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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08 March 2009Instructor: Tasneem Darwish18 Examples 1.10 1)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

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08 March 2009Instructor: Tasneem Darwish19 Examples 1.10 1)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 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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08 March 2009Instructor: Tasneem Darwish20 Examples 1.10 2)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

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08 March 2009Instructor: Tasneem Darwish21 Examples 1.10 2)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

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Chapter 1 Logic and proofs

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