1. 1-6: Deductive Reasoning
Homework 6: p , 17-20
Learning Objectives:
Write condition, converse, inverse and contrapositive statements.
Understand the difference between inductive and deductive reasoning
Apply deductive reasoning techniques
Entry Task:
Draw the next figure in the sequence 1 2
Determine if each conjecture is true. If false, provide a counterexample 3
The quotient of two negative numbers is positive
True 4
The square of an odd integer is an odd integer
True 5
The sum of twice a number and one is always odd
True

2. Concept: Venn Diagrams
Group A
Group B
Groups Overlap
[Intersection]
Groups Don’t Overlap
[Union]
Group A
Group B
Group A
Group B
One group part of other
[Subset]

3. Activity: Building Venn Diagrams
Demonstration on Board

4. Activity: Building Venn Diagrams
Create a three region Venn Diagram comparing three topics of your choice. Fill in each region with the indicated number of elements
Examples: (You Can’t Use)
Call of Duty, Halo, Battlefield
Ford, Chevy, Dodge
Any NFL, NBA, MLB, (or NHL) 1 3 5
You will be graded on:
Quality + Color
Completeness (1,3,5)
Accuracy (Are the elements correct?)
Logic (Does it follow the rules?)

5. Concept: Conditional Statement

6. Example 1: Hypothesis and Conclusion
Identify the hypothesis and conclusion of each conditional.
If today is Thanksgiving Day, then today is Thursday.
Hypothesis: Today is Thanksgiving Day.
Conclusion: Today is Thursday.
A number is a rational number if it is an integer.
Hypothesis: A number is an integer.
Conclusion: The number is a rational number.

7. Student Led Example 1: Hypothesis and Conclusion
Identify the hypothesis and conclusion of the statement.
"A number is divisible by 3 if it is divisible by 6."
Hypothesis: A number is divisible by 6.
Conclusion: A number is divisible by 3.

8. Example 2: Conditional Statements
Write a conditional statement from the following.
An obtuse triangle has exactly one obtuse angle.
Conditional Statements are written
“If [statement 1] then [statement 2]”
If a triangle is obtuse, then it has exactly one obtuse angle.

9. Student Led Example 2: Conditional Statements
Write a conditional statement from the following.
Two angles that are complementary are acute
If two angles are complementary, then they are acute.
A segment bisector creates a midpoint
If a segment is bisected, then it has a midpoint

10. Concept: Other Forms of Statements
Conditional Statement: 𝑝→𝑞
Converse Statement: 𝑞→𝑝
Inverse Statement: ¬𝑝→¬𝑞
Contrapositive: ¬𝑞→¬𝑝

11. Example 3: Types of Statements
𝑝: A triangle is equilateral 𝑞: A triangle has three congruent sides
Conditional: 𝒑→𝒒
If a triangle is equilateral, then it has three congruent sides
Converse: 𝒒→𝒑
If a triangle has three congruent sides, then it’s equilateral

12. Example 3: Types of Statements
𝑝: A triangle is equilateral 𝑞: A triangle has three congruent sides
Inverse: ¬𝒑→¬𝒒
If a triangle is NOT equilateral, then it does NOT have three congruent sides
Contrapositive: ¬𝒒→¬𝒑
If a triangle does NOT have three congruent sides, then it is NOT equilateral

13. Student Led Example 3: Types of Statements
Write the four statement types for the following 𝑝: It’s a quadrilateral 𝑞: it has four sides
Conditional 𝒑→𝒒 :
If it’s a quadrilateral,
then it has four sides
Converse 𝒒→𝒑 :
If it has four sides,
then it’s a quadrilateral
Inverse ¬𝒑→¬𝒒 :
If it’s NOT a quadrilateral,
then it does NOT have four sides
Contrapositive ¬𝒒→¬𝒑 :
If it does NOT have four sides,
then it’s NOT a quadrilateral

14. VS Concept Inductive Reasoning Deductive Reasoning
Uses specific examples to make a general “rule”
Finding patterns or stereotypes
10, 20, 30, 40, 50…
Using the pattern, the rule would be add 10 and the next number would be 50
Takes a general rule and uses it to make a more specific example
Drawing conclusions from previous known facts & definitions
Quadrilaterals have four sides therefore, a square is a quadrilateral.

15. Example 4: Using Venn Diagrams and Deductive Reasoning
The Venn diagram shows the number of people who can or cannot attend the May or the June Spanish Club Meetings
How many people can attend the May OR the June meeting?
𝟐𝟓 because 𝟓+𝟔+𝟏𝟒
How many people can attend BOTH the May and the June meetings?
𝟔; overlapping region
How many people are unable to meeting either meeting?
𝟐; outside region
What does the “5” represent?
The number of people who can only make the May meeting

16. SLE 4: Using Venn Diagrams and Deductive Reasoning
The Venn diagram shows the number of graduates last year who did or did not attend their junior or senior prom.
How many graduates attended their senior but not their junior prom?
𝟐𝟓 (right side only)
How many graduates attended their junior and senior proms?
𝟏𝟐𝟑; overlapping region
How many graduates did not attend either of their proms
𝟑𝟕; outside region
How many students graduated last year? Explain.
185; the number 85 only represents the previous year

17. Concept: Biconditional Statements
A biconditional statement is a conjunction of a true conditional and its true converse. If the conditional and converse are true… Biconditionals are written with “If and Only If”

18. Example 5A: Biconditionals
Write a biconditional statement from the following statements, then determine it’s truth value. If false, provide a counterexample 𝑝: a number is even 𝑞: a number is divisible by two
TRUE
𝑝→𝑞
If a number is even, then it’s divisible by two
𝑞→𝑝
If a number is divisible by two, then it’s even
𝑝↔𝑞
A number is even IF AND ONLY IF it’s divisible by two

19. Example 5B: Biconditionals
Write a biconditional statement from the following statements, then determine it’s truth value. If false, provide a counterexample A square is a quadrilateral
Counterexample: A rectangle is a quadrilateral
𝑝→𝑞
If it’s a square, then it’s a quadrilateral
FALSE
𝑞→𝑝
If it’s a quadrilateral, then it’s a square
𝑝↔𝑞
It’s a square if and only if it’s a quadrilateral

20. Example 5: Biconditionals
Write a biconditional statement from the following statements, then determine it’s truth value. If false, provide a counterexample A
𝑝: An angle is less than 90°, 𝑞: the angle is acute
True: An angle is acute if and only if it is less the 90° B
𝑝: It’s Saturday 𝑞: It’s the weekend
False: Sunday is also on the weekend C
An integer is a rational number
False: ½ is rational, but not an integer

21. End of Lesson
Identify the hypothesis and conclusion of each conditional.
1. A triangle with one right angle is a right triangle.
2. All even numbers are divisible by 2.
3. Determine if the statement “If n2 = 144, then
n = 12” is true. If false, give a counterexample.
H: A triangle has one right angle.
C: The triangle is a right triangle.
H: A number is even.
C: The number is divisible by 2.
False; n can also = –12.