Math 240: Transition to Advanced Math

1. Math 240: Transition to Advanced Math • Deductive reasoning: logic is used to draw conclusions based on statements accepted as true. • Thus conclusions are proved to be true, provided assumptions are true. • If results are incorrect, then assumptions need to be modified. • In this course we focus on logic and proof, as opposed to computational methods such as algebra and calculus.

2. Ch 1.1: Propositions & Connectives • Proposition: A sentence that is either true or false. • Examples: Which is a proposition? • Today is Monday. • 2 + 2 = 5 • 2 + x = 5 • What is mathematics? • Newton wore green socks occasionally. • This sentence is false.

3. Connectives • Definitions: Given two propositions P and Q, • The conjunction of P and Q is denoted by P /\ Q, and represents the proposition “P and Q.” P /\ Q is true exactly when P and Q are true. • The disjunction of P and Q is denoted by P \/ Q, and represents the proposition “P or Q.” P \/ Q is true exactly when at least one of P or Q is true. • The negation of P is denoted by ~P, and represents the proposition “not P.” ~P is true exactly when P is false.

4. Examples • Suppose P = “Chickens ride the bus” and Q = “2 > 1” • Find the truth value of the following propositions • P /\ Q • P \/ Q • ~P • ~Q

5. Propositional Forms • The sentence “Chickens ride the bus or 2 >1” is a proposition, while the symbolic representation “P \/ Q” is a propositional form. (Compare counting with algebra) • A propositional form is an expression involving finitely many logical symbols and letters. • The truth value of a propositional form can be found using a truth table.

6. Truth Tables • A truth table must list all possible combinations of truth values of components of propositional form. • Example: Give the truth table for P /\ Q. • Example: Give the truth table for P \/ Q.

7. Truth Tables • Example: Give the truth table for ~P. • Example: Give the truth table for (P /\ Q) \/ ~Q • Example: Give the truth table for P \/ (Q /\ R) • Example: Find the truth value of (P \/ S) /\ (P \/ T), given that P is true while S and T are false.

8. Equivalence • Definition: Two propositions P and Q are equivalent iff they have the same truth table. • Example: P is equivalent to P \/ (P /\ Q) • Example: P is equivalent to ~(~P)

9. Denial • Definition: A denial of a proposition S is any proposition equivalent to ~S. • Example: Suppose P = “4 is an odd number” • ~P = “It is not the case that 4 is an odd number” • Useful denials: • 4 is not odd • 4 is even • The remainder when dividing 4 by 2 is 0 • Example: Cleopatra was an excellent Math 240 student. Explain.

10. Denial • Does ~(P \/ Q) = (~P) \/ (~Q)? • Does ~(P /\ Q) = (~P) /\ (~Q)? • You will find out in the homework!

11. Order of Operations • Use delimiters such as ( ), { }, [ ], in the usual way. Next • First priority: ~ • Second: /\ • Third: \/ • Example • ~P \/ Q = (~P) \/ Q • P \/ Q /\ R = P \/ (Q /\ R) • Left to right priority: • P \/ Q \/ R = (P \/ Q) \/ R • P \/ Q /\ R \/ ~R = ( P \/ [Q /\ R]) \/ (~R) • Parentheses are good, but can get unwieldy.

12. Tautologies • Definition: A tautology is a propositional form that is true for every assignment of truth values of components. • Example: P \/ ~P is a tautology

13. Contradiction • Definition: A contradiction is a propositional form that is false for every assignment of truth values of components. • Example: P /\ ~P is a contradiction

14. Homework • Read Ch 1.1 • Do 7(1,2a-e,i,3a,b,d-g,j,k,4a-c,h,5a-d,6a,b,7,11a,b)