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
T
F
T
F
Q P /\ Q
T
T
T
F
F
F
F
F
P
T
F
T
F
Q P \/ Q
T
T
T
T
F
T
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 odd
4 is even
The remainder when dividing 4 by 2 is 0
Example: 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. 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.
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)