1. Warm Up Challenge
1. The complement of an angle is 10 less than half the measure of the supplement of the angle. What is the measure of the supplement of the angle? 2. What conclusion can be drawn from the following? ~𝑐→~𝑓 𝑔→𝑏 𝑝→𝑓 𝑐→~𝑏

2. Warm Up Challenge
1. The complement of an angle is 10 less than half the measure of the supplement of the angle. What is the measure of the supplement of the angle? x=160 degrees 2. What conclusion can be drawn from the following? ~𝑐→~𝑓 𝑔→𝑏 𝑝→𝑓 𝑐→~𝑏 𝑝→~𝑔 or g→~𝑝

3. Venn Diagrams and Logic
Chapter 2

4. Using Venn Diagrams
A Venn Diagram is a tool for determining whether a conditional statement is true or false.

5. Example: “If Jenny lives in Oswego, then Jenny lives in Illinois.”
Consider the converse,
“If Jenny lives in Illinois, then
she must live in Oswego”
Is the converse true or false?
Consider the contrapositive,
“If Jenny doesn’t live in Illinois, then she doesn’t live in Oswego.”
Is the contrapositive true or false?

6. Theorem
If the conditional statement is true, then the contrapositive of the statement is also true.
(If p, then q ↔ If ~p then ~q)

7. Brain Break!
Stand Up
Put your arms out in front of you and match your fingers from each hand together and then match your thumbs together.
Now put lower your middle fingers so that the knuckles touch. Keep them flat against each other.
Now un-touch and retouch your thumbs.
Now un-touch and retouch your index fingers.
Now un-touch and retouch your pinkies.
Lastly, un-touch and retouch your ring fingers.

8. Logic
Negation: A statement that has the opposite meaning and truth value of an original statement.
Symbols: ~p, read “not p”

9. Logic
Conjunction: a compound statement formed by joining two or more statements using the word “and”.
Symbols: 𝑝⋀𝑞, read “p and q”

10. Disconjunction: a compound statement formed by joining two or more statements using the word or.
Symbols:𝑝⋁𝑞, read “p or q”

11. Truth Tables
Truth tables can be used to determine truth values of negations and compound statements.

12. Truth Tables Construct a truth table for ~p⋁q
Step 1: Make columns with the heading p, q, ~p and ~p⋁q
Step 2: List the possible combinations of truth values for p and q.
Step 3: Use the truth values of p to determine the truth values of ~p.
Step 4: Use the truth values of ~p and q to write the truth values for ~p⋁q

13. Example 2
Construct a truth table for 𝑝⋁(~𝑞⋀𝑟)

14. Conjunctions can be illustrated with Venn Diagrams.

15. Example 3 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?
B) How many students are enrolled in tap or ballet?
C) How many students are enrolled in jazz and ballet, but not tap?

16. Example 4 The 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?
B) How many students have dogs or cats?
C) How many students have dogs, cats, and birds as pets?