IGE104:
LOGIC AND MATHEMATICS FOR
DAILY LIVING
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Lecture 4: Truth Values
IGE104 3/2010- Lecture 4
March 29, 2011
PROPOSITIONS
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Negation and truth values
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PROPOSITION
Definition: has the structure of a complete sentence and
makes a claim
 “Kate is not sitting in the chair.”



“Eva enjoyed the pizza.”


The claim can be an assertion
“The sky is brown.”


The claim can be a denial
The claim can be either true or false
“What time is it?”

Not a proposition since it is in the form of a question
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NEGATION (OPPOSITES)
The opposite of a proposition is a negation
 The proposition: “the sky is blue” has a negation “the
sky is not blue”


If a proposition is represented as p then the negation is
written as not p or ~p.
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TRUTH VALUES

A proposition makes some claim. Thus it can be either
true or false.

We say a proposition has a truth value of T (true) or F (false)
If a proposition is True then its negation is False
 If a proposition is False then its negation is True

p
T
F

not p
F
T
What are two example propositions?
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DOUBLE NEGATION
“I cannot say that I do not disagree with you”
– Groucho Marx
 not p has the opposite truth value of p, so the double
negation not not p has the opposite truth value of not p
and therefore has the same truth value as p
p
T
F
not p
F
T
not not p
T
F
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EXAMPLE
After reviewing data showing an association between lowlevel radiation and cancer among older workers at the Oak
Ridge National Laboratory, a health scientist from the
University of North Carolina (Chapel Hill) was asked about
the possibility of a similar association among younger
workers at another national laboratory. He was quoted as
saying (Boulder Daily Camera):
My opinion is that it’s unlikely that there is no
association.
Does the scientist think there is an association between
low level radiation and cancer?
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LOGICAL CONNECTORS
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Combining propositions
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LOGICAL CONNECTOR: AND

We can join two propositions with and to get a new
proposition.
Consider two propositions:
 p = I wore my nice blouse
 q = It rained today


The conjunction: I wore my nice blouse and it rained
today

Is true only if both p and q are true
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TRUTH TABLE FOR CONJUNCTIONS (AND)

We have to analyze all possible combinations of the
propositions p as well as q
p
q
p and q
T
T
T
F
T
F
F
T
F
F
F
F
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FIND THE TRUTH VALUE FOR PROPOSITIONS WITH
AND

Are the following propositions true or false?

The capital of France is Paris and Antarctica is cold.

The capital of France is Paris and the capital of Thailand
is Madrid.
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TRIPLE CONJUNCTION

Consider the conjunction p and q and r
p
q
r
T
T
T
T
F
F
T
T
T
F
T
T
F
F
F
T
F
T
F
T
F
F
F
F
IGE104 3/2010- Lecture 4
p and q and r
When is this position
true?
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LOGICAL CONNECTOR: OR

Two types of or
Inclusive or means “either or both”
 Exclusive or means “one or the other, but not both”

An insurance policy can say that it covers hospitalization
in cases of illness or injury (inclusive)
 A restaurant menu gives the choice of soup or salad
(exclusive)


In logic, unless stated otherwise, the or is inclusive.
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OR STATEMENTS (DISJUNCTIONS)

Given two propositions, p , q, the statement p or q is
called their disjunction.
p or q is true when either p is true, q is true, or both
are true.
 p or q is false when both p and q are false.

IGE104 3/2010- Lecture 4
p
q
p or q
T
T
T
F
T
F
F
T
F
T
F
T
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DISJUNCTIONS

Are the following statements true or false?

“airplanes can fly or cows can read”
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IF… THEN STATEMENTS (CONDITIONALS)

Conditional propositions: propose something to be true
based on something else being true
Example of if p then q:
 “If I am elected, then the minimum wage will increase”
p = I am elected
 q = the minimum wage will increase

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CONDITIONAL PROPOSITIONS
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TRUTH VALUES OF IF… THEN STATEMENTS
Example of if p then q:
 “If I am elected, then the minimum wage will increase”
 p = I am elected
 q = the minimum wage will increase
What if both p and q are true?
 What if p is true and q is false?
 What if p is false and q is true?
 What if both p and q are false?

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TRUTH VALUES OF IF… THEN STATEMENTS


We can make a truth table for the conditional statement
if p then q.
p
q
If p then q
T
T
T
F
T
F
F
T
F
T
T
F
Example:
“If you study hard then you will pass the test.”
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REALITY CHECK
What if instead the candidate says:
 “If I am elected, I will personally eliminate all poverty on
Earth.”

According to logic rules, if the candidate is NOT elected
this statement is true.
 Would you consider this true?

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ALTERNATIVE PHRASING OF CONDITIONALS

We can rewrite some phrases in the if…then structure.
Examples:
 “A rise in sea level will devastate Florida”
 “more rain will lead to a flood”
 “Being male is necessary for being president of the
United States”

p is sufficient for q
 p implies q
 q if p

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p will lead to q
q is necessary for p
q whenever p
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THE CONVERSE

The order of the proposition does not matter in a
conjunction or a disjunction.
p and q is the same as q and p
 p or q is the same as q or p

If we switch the order in a conditional statement, then
we get a different proposition, called the converse
 Conditional:
if p then q
“If you are alive then you breathing.”
 Converse:
if q then p
“If you are breathing then you are alive.”

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INVERSE AND CONTRAPOSITIVE

Conditional:

Converse:

Inverse:
if not p, then not q
“If you are not alive then you are not breathing.”

Contrapositive:
if not q, then not p
“If you are not breathing then you are not alive.”
IGE104 3/2010- Lecture 4
if p then q
“If you are alive then you are breathing.”
if q then p
“If you are breathing then you are alive.”
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CONDITIONAL, CONVERSE, INVERSE,
CONTRAPOSITIVE

Here is the truth table for the statements in the previous
slide
p
q
if p then q
T
T
T
F
F
if not p
if q then p then not q
if not q
then not p
(converse)
(inverse)
(contrapositive)
T
T
T
T
F
T
F
T
T
F
T
F
F
T
F
T
T
T
T
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EXAMPLE

“If a creature is a whale then it is a mammal”

Converse:

Inverse:

Contrapositive:
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CLASS ACTIVITY
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10 minutes
IGE104 3/2010- Lecture 4
March 29, 2011
BIBLIOGRAPHY
(1) Bennett, J. O. , and W. L. Briggs. General Education
Mathematics: New Approaches for a New Millennium.
AMATYC Review, vol. 21, no. 1, Fall 1999, pp. 3-16.

http://math.ucdenver.edu/~wbriggs/qr/AMATYCPaper.html
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