1. 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

2. 08 March 2009Instructor: Tasneem Darwish2 Outlines Logical Equivalence. Logical implications. The algebra of Propositions. More about conditionals. Arguments.

3. 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

4. 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

5. 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

6. 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

7. 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

8. 08 March 2009Instructor: Tasneem Darwish8 If we have ‘P ├ Q’ then ‘P → Q’ is a tautology and vice versa. Logical Implication

9. 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

10. 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

11. 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

12. 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

13. 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

14. 08 March 2009Instructor: Tasneem Darwish14 Example 1.8 Prove that The Algebra of Propositions

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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

21. 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

22. 08 March 2009Instructor: Tasneem Darwish22