Splash Screen

1
Splash Screen
2
Lesson Menu
Five-Minute Check (over Lesson 21) 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
3
5-Minute Check 1
Make a conjecture about the next item in the
sequence.1, 4, 9, 16, 25
A. 30 B. 34 C. 36 D. 40
4
5-Minute Check 2
5
5-Minute Check 3
Determine whether the conjecture is true or
false. Given ?ABC, if m?A 60, m?B 60, and
m?C 60. Conjecture ?ABC is an equilateral
triangle.
A. true B. false
6
5-Minute Check 4
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
7
5-Minute Check 5
8
5-Minute Check 6
Find the next two terms in the sequence 243, 81,
27, 9, ....
A. 3, 1 B. 3, 1 C. 3, 1 D. 3, 1
9
Then/Now
You found counterexamples for false conjectures.

• Determine truth values of negations,
  conjunctions, and disjunctions, and represent
  them using Venn diagrams.

• Find counterexamples.

10
Vocabulary

• statement

• truth value
• negation
• compound statement
• conjunction
• disjunction
• truth table

11

• Statement - A statement is a sentence that is
  either true or false. Statements are represented
  using a letter as p or q.
• Truth Value The truth value of a statement is
  either true (T) or false (F).
• p A rectangle is a quadrilateral.
• Truth value - T

12

• Compound statement - Several statements can be
  joined using the word and or or to create a
  compound statement.
• Conjunction - Statement p and statement q joined
  by the word and is a conjunction.
• Symbol p ? q (Read p and q)
• The conjunction p ? q is true only when both p
  and q are true.

13

• Disjunction - Statement p and statement q joined
  by the word or is a disjunction.
• Symbol p ? q (Read p or q)
• The disjunction p ? q is true if p is true, if q
  is true, or if both are true.

14

• Negation not p is the negation of the statement
  p.
• Symbol p (Read not p)
• The statements p and p have opposite truth
  values.

15

• Truth Tables One way to organize the truth
  values of statements is in a truth table.
• NEGATION

P p
T F
F T
16

• CONJUNCTION

p q p q
T T T
T F F
F T F
F F F
17

• DISJUNCTION

p q p v q
T T T
T F T
F T T
F F F
18

• 2.2 LOGIC
• HW pg 103, 11-22, 25-30,
• Pg 105, 35.

19
Example 1
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.
20
Example 1
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.
21
Example 1
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.
22
Example 1
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.
23
Example 2
Truth Values of Disjunctions
24
Example 2
Truth Values of Disjunctions
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.
25
Example 2
Truth Values of Disjunctions
26
Example 2
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 legsr 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.
27
Example 2
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.
28
Example 2
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.
29
Concept
30
Example 3
Construct Truth Tables
A. Construct a truth table for p ? q.
Step 1 Make columns with the heading p, q, p,
and p ? q.
31
Example 3
Construct Truth Tables
A. Construct a truth table for p ? q.
Step 2 List the possible combinations of truth
values for p and q.
32
Example 3
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.
33
Example 3
Construct Truth Tables
A. Construct a truth table for p ? q.
Step 4 Use the truth values of p and q to write
the truth values for p ? q.
Answer
34
Example 3
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).
35
Example 3
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.
36
Example 3
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.
37
Example 3
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.
38
Example 3
Construct Truth Tables
B. Construct a truth table for p ? (q ? r).
Step 5 Use the truth values for q ? r and p to
write the truth values for p ? (q ? r).
Answer
39
Example 3
A. Which sequence of Ts and Fs would
correctlycomplete the last columnof the truth
table for the given compound statement? (p ? q) ?
(q ? r)
A. T B. T C. T D. TF F F FF T F TF F
F FT T F TF F F FT T F FF F F F
40
Example 3
B. Which sequence of Ts and Fs would
correctlycomplete the last columnof the truth
table for the given compound statement? (p ? q) ?
(q ? r)
A. T B. T C. T D. TT T F TT T T TF T
F FT T T TF T F TT T F TF F F F
41
Example 4
Use Venn Diagrams
DANCING The Venn diagram shows the number of
students enrolled in Moniques 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.
42
Example 4
Use Venn Diagrams
DANCING The Venn diagram shows the number of
students enrolled in Moniques 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.
43
Example 4
Use Venn Diagrams
DANCING The Venn diagram shows the number of
students enrolled in Moniques 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.
44
Example 4
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
45
Example 4
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
46
Example 4
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
47
End of the Lesson