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.

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.

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.

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

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.

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.

P Q P /\ Q

T T T

F T F

T F F

F F F

P Q P \/ Q

T T T

F T T

T F T

F F F

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.

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)

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 odd4 is evenThe remainder when dividing 4 by 2 is 0Example: Cleopatra was an excellent Math 240 student. Explain.

Denial

Does ~(P \/ Q) = (~P) \/ (~Q)?

Does ~(P /\ Q) = (~P) /\ (~Q)?

You will find out in the homework!

Order of Operations

Use delimiters such as ( ), { }, [ ], in the usual way. NextFirst priority: ~Second: /\Third: \/Example

~P \/ Q = (~P) \/ QP \/ Q /\ R = P \/ (Q /\ R)

Left to right priority:P \/ Q \/ R = (P \/ Q) \/ RP \/ Q /\ R \/ ~R = ( P \/ [Q /\ R]) \/ (~R)

Parentheses are good, but can get unwieldy.

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

P ~P P \/ ~P

T F T

F T T

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

P ~P P /\ ~P

T F F

F T F

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)