2.2 - Logic
Getting Started
1.) Determine if the conjecture is true or false. If false provide a
counterexample.
If the area of a rectangle is 20m
rectangle are 4m x 5m.
2
then the dimensions of the
2.) Given unlimited water, a 3-Gallon bucket and a 5-Gallon bucket. Your
objective is to get exactly 4 gallons into one of the buckets.
Use the empty space of your sheet to write, draw or some way explain how
you would do this.
3.) Make a conjecture about the next three numbers
1, 1, 2, 3, 5, 8, 13...
2.2 - Logic
New Vocabulary
*Statement
*Compound Statement
*Truth Value
*Conjunction (and)
*Negation
*Disjunction (or)
*Truth Table
2.2 - Logic
New Vocabulary
A STATEMENT is just that... a statement. Its TRUTH VALUE
will be true (T) or false (F). We often represent statements as
p q r.
p: A rectangle is a quadrilateral
Truth Value: (T)
A NEGATION of a statement has the OPPOSITE meaning, as
well as opposite truth value. For example, the negation of the
statement above is "not p" or ~p.
~p: A rectangle is not a quadrilateral
Truth Value: F
A CONDITIONAL STATEMENT is written as "If... Then..." It
also has a TRUTH VALUE. It is written as p --> q
p --> q: If a figure is a rectangle, then it is a quadrilateral.
Truth Value: T
Very, Very, Very Important Terms / Symbols
TERM
SYMBOL
DEFINITION
and
^
Both have to be true.
or
V
One or the other have to be
true
Negation
~
The opposite truth value
Conditional
p --> q
"If... Then"
Converse
q --> p
"If... Then" with hypothesis
and conclusion reversed
Inverse
~p --> ~q
Both hypothesis and
conclusion are negated
~q --> ~p
Original hypothesis and
conclusion are negated and
reversed
pq or iff
p <--> q
When a conditional and its
converse are true, it can be
stated as "If and only if..."
AKA - Tautology
Contrapositive
Biconditional
Guided Practice:
Determine the truth value of p. Then write ~p.
p: A cube is a regular solid.
Truth Value:
~p:
p: It is raining outside
Truth Value:
~p:
2.2 - Logic
New Vocabulary
Two or more statements joined by the word AND or OR form a
COMPOUND STATEMENT . There are two types of compound
statements - CONjunctions and DISjunctions
CONJUNCTIONS
*use AND
*Both parts must be true
DISJUNCTIONS
*use OR
*Only one part
e
*written p
and
q
*written p
or
q
Related Conditional Statements
Conditional Statement
If m<A = 99o , then <A is obtuse.
Truth Value
T
Converse - Switch Hyp. & Conc.
If <A is obtuse, then m<A = 99o .
F
Inverse - Negate Hyp. & Conc.
If m<A ≠ 99o , then <A is not obtuse.
F
Contrapositive - Negate and Switch
If <A is not obtuse, then m<A ≠ 99o .
T
2.2 - Logic
Truth Values of Conjunctions
Use the following statements and diagram to write a
compound statement for each conjunction. Then find its truth
value. Explain your reasoning.
p: The figure is a triangle
q: The figure has two congruent sides
r: The figure has three acute angles
a.) p and r
b.) q
~r
2.2 - Logic
Truth Values of Conjunctions
p: The figure is a triangle
q: The figure has two congruent sides
r: The figure has three acute angles
a.) p and r
b.) q
~r
On your own:
c.) p
q
d.) ~p and ~r
2.2/ 2.3 - Logic - Getting Started
Truth Values of Disjunctions
Name: ________________
Use the following statements to write a compound statement
for each disjunction. Then find its truth value. Explain your
reasoning.
p: February is a fall month
q: February has 30 days
r: Valentine's Day is in February
a.) q or r
b.) p
q
2.2/ 2.3 - Logic - Getting Started
Truth Values of Disjunctions
c.) ~p
r
Name: ________________
Use the following statements to write a compound statement
for each disjunction. Then find its truth value. Explain your
reasoning.
p: February is a fall month
q: February has 30 days
r: Valentine's Day is in February
a.) q or r
b.) p
q
c.) ~p
r
2.2 - Logic
Truth Tables
Truth tables can be used to determine truth values of negations
and compound statements without having to write them out.
Conditional
Negation
p
~p
p
q
p­>q
T
F
T
T
T
F
T
T
F
F
F
T
T
F
F
T
Disjunction
Conjunction
p
q
pVq
p
q
p^q
T
T
T
T
T
T
T
F
T
T
F
F
F
T
T
F
T
F
F
F
F
F
F
F
2.2 - Logic
Construct a Truth Table
Construct a Truth Table for ~p V q.
You will need this slide for your practice fo' sho'!!
1.) Make columns with headings for each original statement,
any negations of the statement, parenthesized compounds and
the compound statement itself.
2.) List possible combinations of truth values.
3.) Use the truth values of p to determine the truth of
negation.
4.) Use truth values for each part of compound statement to
determine truth of the statement.
2.2 ­ Logic
Construct a Truth Table
>
Construct a truth table for ~p q
p
q
T
T
T
F
F
F
F
T
Logical Equivalency:
When two or more statements have the exact same set of outcomes.
p
q
(p^q)
~p
~q
~p V ~q
~(p^q)
2.2 ­ Logic
Venn Diagrams
Conjunctions and Disjunctions can be illustrated with a Venn Diagram. Consider...
p and q: A rectangle is a quadrilateral and rectangle is convex.
quadrilaterals
p
p
p
q
q
convex
q
2.2 - Logic
Reading a Venn Diagram
The Venn Diagram shows
the number of graduates who
did or did not attend their
Junior or Senior Prom.
a.) How many attended their
Senior but not Junior Prom?
b.) How many attended both?
c.) How many didn't go to
either?
Prom Attendance
S
85
123
J
25
37
Name: ______________________________
2.2 - 2.3: Logic, Truth Tables and
Conditional Statements
Date: ____________
Class Period: ______
1.) For the following p and q, write the conditional, converse, inverse and contrapositive. At the end
of each statement, give the truth value. Assume p -> q is (T).
p: If you are an elephant
q: then you do not forget
Conditional: _______________________________________________________________
Converse: _________________________________________________________________
Inverse: __________________________________________________________________
Contrapositive: _____________________________________________________________
2.) Complete the truth tables below.
a.)
b.)
p
q
p^q
(p ^ q) -> p
c.)
p
q
~p
p V ~p
q -> (p V ~p)
d.) Construct a truth table for the following statement:
(p -> q) V ~(q -> p)
p
q
p -> q
p V (q -> p)
p
3.)
a.) Translate the statements into symbolic
form.
b.) Prove that the sentences below are / are
not logically equivalent.
p: "I eat the right foods"
q: "I get sick"
Sentences:
If I eat the right foods, then I don't get
sick.
I don't get sick or I don't eat the right
foods.
q
p
(p -> q)
q
2.2 - 2.3: Logic, Truth Tables and
Conditional Statements
Name: ______________________________
Date: ____________
Class Period: ______
4.) Bailey has switches at the top and bottom of her stairs to control the light in the stairwell. She
notices that when the upstairs switch is up and the downstairs switch is down, then the lights are on.
a.) Complete the truth table to the right.
b1.) If both switches are up, will the light be on?
b2.) Explain.
Position of Switch
Upstairs
Lights on
Downstairs
c.) If the upstairs switch is down,
and the downstairs switch is up,
will the light be on?
5.) Use the Venn Diagram at the right for exercises 5a. - 5e.
5a.) How many kids used only an MP3 player AND DVR?
5b.) How many said they used all 3 types of electronics?
5c.) How many used only a cell phone?
5d.) How many did not have a DVR?
5e.) If there were 360 kids surveyed,
how many didn't use any of the 3?
DVR
MP3
80
30
50
20
40
30
110
Cell
Phone
Name: ______________________________
2.2 - 2.3: Logic, Truth Tables and
Conditional Statements
Date: ____________
Class Period: ______
1.) For the following p and q, write the conditional, converse, inverse and contrapositive. At the end
of each statement, give the truth value. Assume p -> q is (T).
p: If you are an elephant
q: then you do not forget
Conditional: IF you are an elephant, THEN you never forget (T)
Converse: IF you never forget, THEN you are an elephant (F)
Inverse: IF you are not an elephant, THEN you forget (F)
Contrapositive: IF you forget, THEN you are not an elephant (T)
2.) Complete the truth tables below.
a.)
b.)
p
q
p ^ q (p ^ q) -> p
T
T
T
T
F
F
F
p
q
~p
p V ~p
q -> (p V ~p)
T
T
T
F
T
T
F
F
T
F
F
T
T
T
F
F
F
T
T
T
T
F
F
T
F
F
T
T
T
c.)
d.) Construct a truth table for the following statement:
(p -> q) V ~(q -> p)
p
q
p -> q p V (q -> p)
T
T
T
T
F
F
F
p
q
(p -> q)
(q->p)
~(q->p) (p->q)V~(p->q)
T
T
T
T
T
F
T
F
T
T
F
F
T
F
F
T
T
F
F
T
T
F
T
T
F
T
T
F
F
T
T
F
T
3.)
a.) Translate the statements into symbolic
form.
b.) Prove that the sentences below are / are
not logically equivalent.
p: "I eat the right foods"
q: "I get sick"
p
q
~p
~q
p -> ~q
~q V ~p
T
T
F
F
F
F
T
F
F
T
T
T
F
T
T
F
T
T
Sentences:
If I eat the right foods, then I don't get
sick. p -> ~q
I don't get sick or I don't eat the right
foods. ~q V ~p
F
F
T
T
T
T
2.2 - 2.3: Logic, Truth Tables and
Conditional Statements
Name: ______________________________
Date: ____________
Class Period: ______
4.) Bailey has switches at the top and bottom of her stairs to control the light in the stairwell. She
notices that when the upstairs switch is up and the downstairs switch is down, then the lights are on.
a.) Complete the truth table to the right.
b1.) If both switches are up, will the light be on? NO
b2.) Explain. To have both switches up would be one
movement turning the lights off.
Position of Switch
Lights on
Upstairs
Downstairs
Up
Down
On
Up
Up
Off
Down
Down
On
Down
Up
On
c.) If the upstairs switch is down,
and the downstairs switch is up,
will the light be on? Yes - the switches must be in
opposite positions in order to be on.
5.) Use the Venn Diagram at the right for exercises 5a. - 5e.
5a.) How many kids used only an MP3 player AND DVR? 50
5b.) How many said they used all 3 types of electronics? 40
5c.) How many used only a cell phone? 110
5d.) How many did not have a DVR? 210
5e.) If there were 360 kids surveyed,
how many didn't use any of the 3? 0
DVR
MP3
80
30
50
20
40
30
110
Cell
Phone