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

04 March 2009Instructor: Tasneem Darwish2 Outlines Propositions and Truth Values. Logical Connectives and Truth Tables. Tautologies and Contradictions.

04 March 2009Instructor: Tasneem Darwish3 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 = 4. 6 < 24. Tomorrow is Thursday. Propositions and Truth Values

04 March 2009Instructor: Tasneem Darwish4 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 Values

04 March 2009Instructor: Tasneem Darwish5 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

04 March 2009Instructor: Tasneem Darwish6 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

04 March 2009Instructor: Tasneem Darwish7 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: Logical Connectives and Truth Tables

04 March 2009Instructor: Tasneem Darwish8 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. Logical Connectives and Truth Tables

04 March 2009Instructor: Tasneem Darwish9 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

04 March 2009Instructor: Tasneem Darwish10 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

04 March 2009Instructor: Tasneem Darwish11 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

04 March 2009Instructor: Tasneem Darwish12 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

04 March 2009Instructor: Tasneem Darwish13 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

04 March 2009Instructor: Tasneem Darwish14 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

04 March 2009Instructor: Tasneem Darwish15 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

04 March 2009Instructor: Tasneem Darwish16 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

04 March 2009Instructor: Tasneem Darwish17 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

04 March 2009Instructor: Tasneem Darwish18 Consider the propositions p: ‘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

04 March 2009Instructor: Tasneem Darwish19 Construct truth tables for the following compound propositions. (i) ￣ p ∨ q(ii) ￣ p ∧ ￣ q (iii) ￣ q → p(iv) ￣ p ↔ ￣ q. Examples 1.1

04 March 2009Instructor: Tasneem Darwish20 Construct truth tables for: (i) p →(q ∧ r )(ii) ( ￣ p ∨ q) ↔ ￣ r. Examples 1.1

04 March 2009Instructor: Tasneem Darwish21 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

04 March 2009Instructor: Tasneem Darwish22 Example 1.3: Show that (p ∧ ￣ q) ∧ ( ￣ p ∨ q) is a contradiction Tautologies and Contradictions

04 March 2009Instructor: Tasneem Darwish23