Transcript
SECTION 2-2Logic
Thursday, October 27, 2011
Essential Questions
How do you determine truth values of negatives, conjunctions, disjunctions, and represent them using Venn diagrams?
Thursday, October 27, 2011
Vocabulary1. Statement:
2. Truth Value:
3. Negation:
4. Compound Statement:
5. Conjunction:
6. Disjunction:
7. Truth Table:
Thursday, October 27, 2011
Vocabulary1. Statement: A sentence that is either true or false, often represented
as statements p and q
2. Truth Value:
3. Negation:
4. Compound Statement:
5. Conjunction:
6. Disjunction:
7. Truth Table:
Thursday, October 27, 2011
Vocabulary1. Statement: A sentence that is either true or false, often represented
as statements p and q
2. Truth Value: The statement is either true (T) or false (F)
3. Negation:
4. Compound Statement:
5. Conjunction:
6. Disjunction:
7. Truth Table:
Thursday, October 27, 2011
Vocabulary1. Statement: A sentence that is either true or false, often represented
as statements p and q
2. Truth Value: The statement is either true (T) or false (F)
3. Negation: A statement with the opposite meaning and opposite truth value; the negation of p is ~p
4. Compound Statement:
5. Conjunction:
6. Disjunction:
7. Truth Table:
Thursday, October 27, 2011
Vocabulary1. Statement: A sentence that is either true or false, often represented
as statements p and q
2. Truth Value: The statement is either true (T) or false (F)
3. Negation: A statement with the opposite meaning and opposite truth value; the negation of p is ~p
4. Compound Statement: Two or more statements joined by and or or
5. Conjunction:
6. Disjunction:
7. Truth Table:
Thursday, October 27, 2011
Vocabulary1. Statement: A sentence that is either true or false, often represented
as statements p and q
2. Truth Value: The statement is either true (T) or false (F)
3. Negation: A statement with the opposite meaning and opposite truth value; the negation of p is ~p
4. Compound Statement: Two or more statements joined by and or or
5. Conjunction:
6. Disjunction:
7. Truth Table:
A compound statement using the word and or symbol ∧
Thursday, October 27, 2011
Vocabulary1. Statement: A sentence that is either true or false, often represented
as statements p and q
2. Truth Value: The statement is either true (T) or false (F)
3. Negation: A statement with the opposite meaning and opposite truth value; the negation of p is ~p
4. Compound Statement: Two or more statements joined by and or or
5. Conjunction:
6. Disjunction:
7. Truth Table:
A compound statement using the word and or symbol ∧A compound statement using the word or or symbol ∨
Thursday, October 27, 2011
Vocabulary1. Statement: A sentence that is either true or false, often represented
as statements p and q
2. Truth Value: The statement is either true (T) or false (F)
3. Negation: A statement with the opposite meaning and opposite truth value; the negation of p is ~p
4. Compound Statement: Two or more statements joined by and or or
5. Conjunction:
6. Disjunction:
7. Truth Table: A way to organize truth values of statements
A compound statement using the word and or symbol ∧A compound statement using the word or or symbol ∨
Thursday, October 27, 2011
Example 1Use the following statements to write a compound sentence for each
conjunction. Then find its truth value. Explain your reasoning.
p: One meter is 100 mm q: November has 30 days
r: A line is defined by two points
a. p and q
b. ~ p ∧ r
Thursday, October 27, 2011
Example 1Use the following statements to write a compound sentence for each
conjunction. Then find its truth value. Explain your reasoning.
p: One meter is 100 mm q: November has 30 days
r: A line is defined by two points
a. p and q
One meter is 100 mm and November has 30 days
b. ~ p ∧ r
Thursday, October 27, 2011
Example 1Use the following statements to write a compound sentence for each
conjunction. Then find its truth value. Explain your reasoning.
p: One meter is 100 mm q: November has 30 days
r: A line is defined by two points
a. p and q
One meter is 100 mm and November has 30 daysq is true, but p is false, so the conjunction is false
b. ~ p ∧ r
Thursday, October 27, 2011
Example 1Use the following statements to write a compound sentence for each
conjunction. Then find its truth value. Explain your reasoning.
p: One meter is 100 mm q: November has 30 days
r: A line is defined by two points
a. p and q
One meter is 100 mm and November has 30 daysq is true, but p is false, so the conjunction is false
One meter is not 100 mm and a line is defined by two points
b. ~ p ∧ r
Thursday, October 27, 2011
Example 1Use the following statements to write a compound sentence for each
conjunction. Then find its truth value. Explain your reasoning.
p: One meter is 100 mm q: November has 30 days
r: A line is defined by two points
a. p and q
One meter is 100 mm and November has 30 daysq is true, but p is false, so the conjunction is false
One meter is not 100 mm and a line is defined by two points
b. ~ p ∧ r
Both ~p and r are true, so is true ~ p ∧ r
Thursday, October 27, 2011
Example 2Use the following statements to write a compound statement for each disjunction. Then find its truth value. Explain your reasoning.
q: Kilometers are metric units
r: 15 is a prime number
a. p or q
b. q ∨ r
p: AB is proper notation for “ray AB”
Thursday, October 27, 2011
Example 2Use the following statements to write a compound statement for each disjunction. Then find its truth value. Explain your reasoning.
q: Kilometers are metric units
r: 15 is a prime number
a. p or q
b. q ∨ r
p: AB is proper notation for “ray AB”
AB is proper notation for “ray AB” or kilometers are metric units
Thursday, October 27, 2011
Example 2Use the following statements to write a compound statement for each disjunction. Then find its truth value. Explain your reasoning.
q: Kilometers are metric units
r: 15 is a prime number
a. p or q
Both p and q are true, so p or q is true
b. q ∨ r
p: AB is proper notation for “ray AB”
AB is proper notation for “ray AB” or kilometers are metric units
Thursday, October 27, 2011
Example 2Use the following statements to write a compound statement for each disjunction. Then find its truth value. Explain your reasoning.
q: Kilometers are metric units
r: 15 is a prime number
a. p or q
Both p and q are true, so p or q is true
Kilometers are metric units or 15 is a prime number
b. q ∨ r
p: AB is proper notation for “ray AB”
AB is proper notation for “ray AB” or kilometers are metric units
Thursday, October 27, 2011
Example 2Use the following statements to write a compound statement for each disjunction. Then find its truth value. Explain your reasoning.
q: Kilometers are metric units
r: 15 is a prime number
a. p or q
Both p and q are true, so p or q is true
Kilometers are metric units or 15 is a prime number
b. q ∨ r
Since one of the statements (q) is true, is true q ∨ r
p: AB is proper notation for “ray AB”
AB is proper notation for “ray AB” or kilometers are metric units
Thursday, October 27, 2011
Example 2Use the following statements to write a compound statement for each disjunction. Then find its truth value. Explain your reasoning.
q: Kilometers are metric units
r: 15 is a prime number
c. ~ p ∨ r
p: AB is proper notation for “ray AB”
Thursday, October 27, 2011
Example 2Use the following statements to write a compound statement for each disjunction. Then find its truth value. Explain your reasoning.
q: Kilometers are metric units
r: 15 is a prime number
c. ~ p ∨ r
p: AB is proper notation for “ray AB”
AB is not proper notation of “ray AB” or 15 is a prime number
Thursday, October 27, 2011
Example 2Use the following statements to write a compound statement for each disjunction. Then find its truth value. Explain your reasoning.
q: Kilometers are metric units
r: 15 is a prime number
c. ~ p ∨ r
Since both ~p and r are false, is false ~ p ∨ r
p: AB is proper notation for “ray AB”
AB is not proper notation of “ray AB” or 15 is a prime number
Thursday, October 27, 2011
Example 3Construct a truth table for the following.
a. p ∧ q
Thursday, October 27, 2011
Example 3Construct a truth table for the following.
a. p ∧ q
p q p ∧ q
T T T
T F F
F T F
F F F
Thursday, October 27, 2011
Example 3Construct a truth table for the following.
a. p ∧ q
p q p ∧ q
T T T
T F F
F T F
F F F
Thursday, October 27, 2011
Example 3Construct a truth table for the following.
a. p ∧ q
p q p ∧ q
T T T
T F F
F T F
F F F
Thursday, October 27, 2011
Example 3Construct a truth table for the following.
b. ~ p ∧ (~ p ∨ q)
Thursday, October 27, 2011
Example 3Construct a truth table for the following.
b. ~ p ∧ (~ p ∨ q)
p q ~p ~p∨q ~p∧(~p∨q)
T T F T F
T F F F F
F T T T T
F F T T T
Thursday, October 27, 2011
Example 3Construct a truth table for the following.
b. ~ p ∧ (~ p ∨ q)
p q ~p ~p∨q ~p∧(~p∨q)
T T F T F
T F F F F
F T T T T
F F T T T
Thursday, October 27, 2011
Example 3Construct a truth table for the following.
b. ~ p ∧ (~ p ∨ q)
p q ~p ~p∨q ~p∧(~p∨q)
T T F T F
T F F F F
F T T T T
F F T T T
Thursday, October 27, 2011
Example 3Construct a truth table for the following.
b. ~ p ∧ (~ p ∨ q)
p q ~p ~p∨q ~p∧(~p∨q)
T T F T F
T F F F F
F T T T T
F F T T T
Thursday, October 27, 2011
Example 3Construct a truth table for the following.
b. ~ p ∧ (~ p ∨ q)
p q ~p ~p∨q ~p∧(~p∨q)
T T F T F
T F F F F
F T T T T
F F T T T
Thursday, October 27, 2011
Example 4The Venn diagram shows the number of students enrolled in Maggie
Brann’s Dance School for tap, jazz, and ballet classes.
Tap28
Jazz43
Ballet29
9
2517
25
a. How many students are enrolled in all three classes?
b. How many students are enrolled in tap or ballet?
c. How many students are enrolled in jazz and ballet, but not tap?
Thursday, October 27, 2011
Example 4The Venn diagram shows the number of students enrolled in Maggie
Brann’s Dance School for tap, jazz, and ballet classes.
Tap28
Jazz43
Ballet29
9
2517
25
a. How many students are enrolled in all three classes?
b. How many students are enrolled in tap or ballet?
c. How many students are enrolled in jazz and ballet, but not tap?
9 students
Thursday, October 27, 2011
Example 4The Venn diagram shows the number of students enrolled in Maggie
Brann’s Dance School for tap, jazz, and ballet classes.
Tap28
Jazz43
Ballet29
9
2517
25
a. How many students are enrolled in all three classes?
b. How many students are enrolled in tap or ballet?
c. How many students are enrolled in jazz and ballet, but not tap?
9 students
28 + 25 + 9 + 17 + 29 +25
Thursday, October 27, 2011
Example 4The Venn diagram shows the number of students enrolled in Maggie
Brann’s Dance School for tap, jazz, and ballet classes.
Tap28
Jazz43
Ballet29
9
2517
25
a. How many students are enrolled in all three classes?
b. How many students are enrolled in tap or ballet?
c. How many students are enrolled in jazz and ballet, but not tap?
9 students
28 + 25 + 9 + 17 + 29 +25133 students
Thursday, October 27, 2011
Example 4The Venn diagram shows the number of students enrolled in Maggie
Brann’s Dance School for tap, jazz, and ballet classes.
Tap28
Jazz43
Ballet29
9
2517
25
a. How many students are enrolled in all three classes?
b. How many students are enrolled in tap or ballet?
c. How many students are enrolled in jazz and ballet, but not tap?
9 students
28 + 25 + 9 + 17 + 29 +25133 students
25 students
Thursday, October 27, 2011
Check Your Understanding
Problems 1-10 on page 101 can help you make sure you understand these concepts. Take a look at those problems and see what you can
do.
Thursday, October 27, 2011
Problem Set
Thursday, October 27, 2011
Problem Set
p. 101 #11-32, 41
“Lack of money is no obstacle. Lack of an idea is an obstacle.” - Ken Hakuta
Thursday, October 27, 2011
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