Expressions (Propositional formulas or forms)

1. Expressions (Propositional formulas or forms) Instructor: Hayk Melikya melikyan@nccu.edu

2. Worm_UP Definition:The disjunction of the statements P and Q is the statement "P or Q" and is denoted by P  Q: • P  Q is true if of P and Q true; otherwise, P  Q is false. . P1 : the integer 2 is even. P2 : the integer 34 is prime. P1  P2 : is ----------

3. Worm_Up Definition: The conjunction of the statements P and Q is the statement "P and Q" and is denoted by P ^ Q: • P ^ Q is true if of P and Q true; otherwise, P ^ Q is false. P^Q: is ------------------

4. Exercise 1 • How would you write each of these statement using combinations of P: (meaning "Sue is an English major") and Q: (meaning "Sue is a junior") with the operations , , ~. • 1. Sue is a junior English major. • 2. Sue is either an English major or she is a junior. • 3. Sue is a junior, but she is not an English major. • 4. Sue is neither an English major nor a junior. • 5. Sue is exactly one of the following: an English major or a junior.

5. Implication • There are several ways of expressing P Q : 1. If P, then Q 2. Q if P 3. P implies Q 4. P only if Q (P is true only under the condition that Q is true) i.e., it cannot be the case that P is true and Q is false. Thus, if P is true, then necessarily Q must be true. 5. Q is necessary for P 6. P is sufficient for Q (the truth of P is sufficient for the truth of Q)

6. Exercise 2 Rewrite each of the following sentences in "if, then" form: (a) You will pass the test only if you study for at least four hours. (b) Attending class regularly is a necessary condition for passing the course. (c) In order to be a square, it is sufficient that the quadrilateral have four equal angles. (d) In order to be a square, it is necessary that the quadrilateral have four equal angles. (e) An integer is an odd prime only if it is greater than 22

7. Propositional Expressions (Forms) • Alphabet: variables (propositional variables) (letters X, Y, Z, … A, B, C ) symbols , , ~, and parentheses ( , ) also we add two more ,  • Expressions are formed using these elements of alphabet as follows: 1. Each variable is expression 2. IF X and Y are expressions then ~ X, XY, XY, XY, XY and (X) all are expressions 3. Any expression is obtained by applying repeatedly, steps 1 or 2.

8. Examples: • (XY)Z , (D), ((XY)(~ X Z)) X, are propositional expressions. • (( X Y ((Y)() ,  PP, QR - are not propositiona expressions. • At this point all these expressions have no meaning whatsoever. But if one replaces all the variables in expression (XY)Z by the propositions one will obtain a proposition and as any other proposition it can be evaluated either true or false.

9. Order of Operations • To interpret a propositional expression, read from left to right and use the following order ( precedence): 1. propositional expressions within parentheses ( innermost first) 2. negations, 3. conjunctions, 4. disjunctions, 5. conditionals, 6. biconditionals.

10. Tautology and contradiction Definition: A compound proposition is a proposition composed of one or more given propositions (called the component propositions in this context) and at least one logical connective. Definition: A compound proposition P is called a tautology if it is true for all possible combinations of truth values of the component propositions that compose P Definition: A compound Proposition S is called a contradiction if it is false for all possible combinations of truth values of the component propositions that compose S

11. Very Important Tautologies:, , ~, , ,  • Commutative Law [Com] 2. Associative Law [Assoc] PQ  QP , (PQ ) R P(Q  R) PQ  QP (P  Q )  R P (Q  R) 3. Distributive Law [Dist] 4. Contrapositive Law [Contr] P(Q  R) (PQ)(P R) (PQ) (~Q~P) P (Q R) (PQ)  (P  R) 5. DeMorgan Law [DeM] 6. Double Negation [DN] ~( PQ)  (~P~Q ) ~~(P) P ~( PQ)  (~P~Q ) 7. Implication Law [Impl] 8. Equivalence Law [Equiv] (PQ)(~PQ) PQ  (PQ)  (QP) PQ  (PQ)(~Q  ~P) 9. Exportation [Exp] 10. Tautology [Taut] (PQ)RP(QR) PPP PPP

12. Inference Rules (Valid arguments) Let P1, P2, ..Pk are propositional expressions (PE) . If PE Q is true when all P1, P2, ..Pk are true then we say that Q logically follows from hypotheses P1, P2, ..Pk P1, P2, …,Pk├ Q (├ is called turnstile) That what is called Inference rule or Valid argument Example: P, PQ├ Q P, Q├ PQ

13. Two Methods of Inference rules First: P1, P2, …,Pk├ Q if and only if P1 P2 …  PkQ tautology Example: PQ , P├ Q ( MP) What about PQ , P├ P ??????

14. Second Method To prove that P1, P2, …,Pk├ Q enough to construct sequence of PE Q1, Q2, …, Qn Such thatQn= Q everyQiis either one of Pi ( i = 1, 2, . . . , k) or it follows by the rule of logic

15. Inference Rules , , ~, Let P, Q, R, S are the PE then 1. Modus Ponens [MP] 2. Modus Tolens [MT] PQ , P├ P PQ , ~Q├ ~P 3. Constructive Dilemma [CD] 4. Simplification [Simp] (PQ) (RS), P  R ├ Q  S PQ P 5. Conjunction [Conj] 6. Disjunctive Syllogism [DS] P, Q ├ PQ P  Q , ~P ├ Q 7. Destructive Dilemma [DD] (PQ) (RS), ~Q ~S ├ ~P  ~R 8. Transitivity PQ , QR├ PR

16. Definitions • Two expressions are called (logically) equivalent if they have same truth table for all possible (True or False) values for all variables appearing in either expression. We use the following notation XY to indicate that expressions X and Y are equivalent. • Do not be confused to use symbol  (biconditional) instead of logical equivalence . • We already know that X  Y and , ~ X Y have same truth table so they are logically equivalent therefore (X  Y)(~ XY) • What about X  Y and X  Z (are they equivalent?)

17. Tautology and Contradiction Definition: A propositional expression is called tautology if it yields a true proposition regardless of what propositions replace its variables. • As you know (P  Q)  ( ~Q  ~P) therefore (P  Q) (~Q  ~P) is tautology Definition: A propositional expression is called a contradiction if it yields a False proposition regardless of what propositions replace its variables.

18. Proposition Each of the following propositional expressions is a tautology. • (P  Q) ( ~P Q) • ~(P Q)  (P  ~Q ) • ~(P  Q) ( ~P ~Q) • ~(P  Q) ( ~P~ Q) • ~( ~P)P Proof: Some comments about proposition1.1