Five-Minute Check (over Lesson 2–1)
Then/Now
New Vocabulary
Example 1: Truth Values of Conjunctions
Example 2: Truth Values of Disjunctions
Concept Summary: Negation, Conjunction, Disjunction
Example 3: Construct Truth Tables
Example 4: Real-World Example: Use Venn Diagrams
Over Lesson 2–1
Make a conjecture about the next item in the
sequence.
1, 4, 9, 16, 25
A. 30
B. 34
C. 36
D. 40
Over Lesson 2–1
Make a conjecture about the next item in the
sequence.
A.
B.
C.
D.
Over Lesson 2–1
Determine whether the conjecture is true or false.
Given: ΔABC, if mA = 60, mB = 60, and
mC = 60.
Conjecture: ΔABC is an equilateral triangle.
A. true
B. false
Over Lesson 2–1
Determine whether the conjecture is true or false.
Given: 1 and 2 are supplementary angles.
Conjecture: 1 and 2 are congruent.
A. true
B. false; m1 = 70
and m2 = 110
Over Lesson 2–1
Determine whether the conjecture is true or false.
Given: ΔRST has two congruent sides.
Conjecture:
A. true
B. false;
Over Lesson 2–1
Find the next two terms in the sequence 243, –81,
27, –9, ....
A. –3, –1
B. 3, –1
C. 3, 1
D. –3, 1
You found counterexamples for false
conjectures.
• Determine truth values of negations,
conjunctions, and disjunctions, and
represent them using Venn diagrams.
• Find counterexamples.
• statement
• truth value
• negation
• compound statement
• conjunction
• disjunction
• truth table
Truth Values of Conjunctions
A. Use the following statements to write a compound
statement for the conjunction p and q. Then find its
truth value.
p: One foot is 14 inches.
q: September has 30 days.
r: A plane is defined by three noncollinear points.
Answer: p and q: One foot is 14 inches, and September
has 30 days.
Although q is true, p is false. So, the
conjunction of p and q is false.
Truth Values of Conjunctions
B. Use the following statements to write a compound
statement for the conjunction ~p  r. Then find its
truth value.
p: One foot is 14 inches.
q: September has 30 days.
r: A plane is defined by three noncollinear points.
Answer: ~p  r: A foot is not 14 inches, and a plane is
defined by three noncollinear points.
~p  r is true, because ~p is true and r is true.
A. Use the following statements to write a compound
statement for p and r. Then find its truth value.
p: June is the sixth month of the year.
q: A square has five sides.
r: A turtle is a bird.
A. A square has five sides and a turtle is a
bird; false.
B. June is the sixth month of the year and
a turtle is a bird; true.
C. June is the sixth month of the year and
a square has five sides; false.
D. June is the sixth month of the year and
a turtle is a bird; false.
B. Use the following statements to write a compound
statement for ~q  ~r. Then find its truth value.
p: June is the sixth month of the year.
q: A square has five sides.
r: A turtle is a bird.
A. A square has five sides and a turtle is
not a bird; true.
B. A square does not have five sides and
a turtle is not a bird; true.
C. A square does not have five sides and
a turtle is a bird; false.
D. A turtle is not a bird and June is the
sixth month of the year; true.
Truth Values of Disjunctions
A. Use the following statements to write a compound
statement for the disjunction p or q. Then find its
truth value.
p:
is proper notation for “segment AB.”
q: Centimeters are metric units.
r: 9 is a prime number.
Answer:
is proper notation for “segment AB,” or
centimeters are metric units. Both p and q are
true, so p or q is true.
Truth Values of Disjunctions
B. Use the following statements to write a compound
statement for the disjunction q  r. Then find its truth
value.
p:
is proper notation for “segment AB.”
q: Centimeters are metric units.
r: 9 is a prime number.
Answer: Centimeters are metric units, or 9 is a prime
number. q  r is true because q is true. It does
not matter that r is false.
Truth Values of Disjunctions
C. Use the following statements to write a compound
statement for the disjunction ~p  r. Then find its
truth value.
p:
is proper notation for “segment AB.”
q: Centimeters are metric units.
r: 9 is a prime number.
___
Answer: AB is not proper notation for “segment AB,” or
9 is a prime number. Since not p and r are
both false, ~p  r is false.
A. Use the following statements to write a compound
statement for p or r. Then find its truth value.
p: 6 is an even number.
q: A cow has 12 legs
r: A triangle has 3 sides.
A. 6 is an even number or a cow has
12 legs; true.
B. 6 is an even number or a triangle has
3 sides; true.
C. A cow does not have 12 legs or 6 is an
even number; true.
D. 6 is an even number or a triangle does
not have 3 side; true.
B. Use the following statements to write a compound
statement for ~q  ~r. Then find its truth value.
p: 6 is an even number.
q: A cow has 12 legs.
r: A triangle has 3 sides.
A. A cow does not have 12 legs or a
triangle does not have 3 sides; true.
B. A cow has 12 legs or a triangle has
3 sides; true.
C. 6 is an even number or a triangle has
3 sides; true.
D. A cow does not have 12 legs and a
triangle does not have 3 sides; false.
C. Use the following statements to write a compound
statement for ~p  q. Then find its truth value.
p: 6 is an even number.
q: A cow has 12 legs.
r: A triangle has 3 sides.
A. 6 is an even number or a cow has
12 legs; true.
B. 6 is not an even number or a cow does
not have 12 legs; true.
C. A cow does not have 12 legs, or a
triangle has 3 sides; true.
D. 6 is not an even number or a cow has
12 legs; false.
Construct Truth Tables
A. Construct a truth table for ~p  q.
Step 1
Make columns with the heading p, q, ~p, and
~p  q.
Construct Truth Tables
A. Construct a truth table for ~p  q.
Step 2
List the possible combinations of truth values
for p and q.
Construct Truth Tables
A. Construct a truth table for ~p  q.
Step 3
Use the truth values of p to determine the truth
values of ~p.
Construct Truth Tables
A. Construct a truth table for ~p  q.
Step 4
Answer:
Use the truth values of ~p and q to write the
truth values for ~p  q.
Construct Truth Tables
B. Construct a truth table for p  (~q  r).
Step 1
Make columns with the headings p, q, r, ~q,
~q  r, and p  (~q  r).
Construct Truth Tables
B. Construct a truth table for p  (~q  r).
Step 2
List the possible combinations of truth values
for p, q, and r.
Construct Truth Tables
B. Construct a truth table for p  (~q  r).
Step 3
Use the truth values of q to determine the truth
values of ~q.
Construct Truth Tables
B. Construct a truth table for p  (~q  r).
Step 4
Use the truth values for ~q and r to write the
truth values for ~q  r.
Construct Truth Tables
B. Construct a truth table for p  (~q  r).
Step 5
Answer:
Use the truth values for ~q  r and p to write
the truth values for p  (~q  r).
A. Which sequence of
Ts and Fs would correctly
complete the last column
of the truth table for the
given compound
statement? (p  q)  (q  r)
A. T
F
F
F
T
F
T
F
B. T
F
T
F
T
F
T
F
C. T D. T
F
F
F
T
F
F
F
T
F
F
F
F
F
F
B. Which sequence of
Ts and Fs would correctly
complete the last column
of the truth table for the
given compound
statement? (p  q)  (q  r)
A. T
T
T
F
T
F
T
F
B. T
T
T
T
T
T
T
F
C. T D. T
F
T
T
T
F
F
T
T
F
T
F
T
F
F
Use Venn Diagrams
DANCING The Venn diagram shows the number of
students enrolled in Monique’s Dance School for tap,
jazz, and ballet classes.
A. How many students are enrolled in all three
classes?
The students that are enrolled in
all three classes are represented
by the intersection of all three
sets.
Answer: There are 9 students
enrolled in all three classes.
Use Venn Diagrams
DANCING The Venn diagram shows the number of
students enrolled in Monique’s Dance School for tap,
jazz, and ballet classes.
B. How many students are enrolled in tap or ballet?
The students that are enrolled in
tap or ballet are represented by
the union of these two sets.
Answer: There are 28 + 13 + 9 + 17
+ 25 + 29 or 121 students
enrolled in tap or ballet.
Use Venn Diagrams
DANCING The Venn diagram shows the number of
students enrolled in Monique’s Dance School for tap,
jazz, and ballet classes.
C. How many students are enrolled in jazz and ballet,
but not tap?
The students that are enrolled in
jazz and ballet, but not tap, are
represented by the intersection of
jazz and ballet minus any students
enrolled in tap.
Answer: There are 25 + 9 – 9 or 25
students enrolled in jazz
and ballet, but not tap.
PETS The Venn diagram shows
the number of students at
Manhattan School that have dogs,
cats, and birds as household pets.
A. How many students in
Manhattan School have a dog,
a cat, or a bird?
A. 226
B. 311
C. 301
D. 110
Pets
PETS The Venn diagram shows
the number of students at
Manhattan School that have dogs,
cats, and birds as household pets.
B. How many students have dogs
or cats?
A. 57
B. 242
C. 252
D. 280
Pets
PETS The Venn diagram shows
the number of students at
Manhattan School that have dogs,
cats, and birds as household pets.
C. How many students have dogs,
cats, and birds as pets?
A. 10
B. 85
C. 116
D. 311
Pets