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A =poodle ... a dog

...B dog

.A

B= horse ... NOT a dog

.

B

Venn diagrams:- show relationships between different sets of data.
- can represent conditional statements.
- is usually drawn as a circle.
- Every point IN the circle belongs to that set.
- Every point OUT of the circle does not.

Example:

Lesson 2-2: Logic

For all..., every..., if...then...

Example1:

All right angles are congruent.

Congruent Angles

Example 2:

Every rose is a flower.

Right Angles

Flower

- lines that do
- not intersect

Rose

parallel lines

Example 3:

If two lines are parallel, then they do not intersect.

Lesson 2-2: Logic

To Show Relationships using Venn Diagrams:

Blue or Brown (includes Purple) … AB

A

B

A B

Lesson 2-2: Logic

Example:

Twenty-four members of Mu Alpha Theta went to a Mathematics conference.

One-third of the members ran cross country.

One sixth of the members were on the football team .

Three members were on cross country and football teams.

The rest of the members were in the band.

How many were in the band?

Hint:

Draw a Venn Diagram and take one sentence at a time...

Lesson 2-2: Logic

Solution:

Twenty-four members of Mu Alpha Theta went to a Mathematics conference.

- Three members were on cross country and football teams…

CC

Football

3

1

5

The above sentence tells you two draw overlapping circles and put 3 in CCF

- One-third of the members ran cross country.

24 / 3 = 8; 8 members run cross country. So put 5 in cross country as there are already 3 in cross country.

- One sixth of the members were on the football team .

24/6 = 4; 4 members play football. So put 1 in football as there are already 3 in football.

Continued….

Lesson 2-2: Logic

Example: Continued……

- The rest of the members were in the band. How many were in the band?
- Out of 24 members in Mu Alpha Theta, 9 play football or run cross country.
- Therefore, 15 are in the band.

CC

Football

Band

3

5

1

15

Mu Alpha Theta

Lesson 2-2: Logic

Law of Detachment

Given:

a true conditional statement and

the hypothesis occurs

pq is true

p is given

Conclusion:

the conclusion will also occur

q is true

Lesson 2-2: Logic

Law of Detachment - Example

Example 1:

Given:If three points are collinear, then the points are all on one line.

E, F, and G are collinear.

Conclusion:E, F, and G are all on one line.

Example 2:

Given: If I find $20 in the street, then I’ll take you to the movies.

On October 10 I found $20 in the street.

Conclusion:I will take you to the movies.

Lesson 2-2: Logic

Law of Syllogism

Given:

Two true conditional statements and

the conclusion of the first is the hypothesis of the second.

pq and qr

Conclusion:

If the hypothesis of the first occurs, then the conclusion of the second will also occur.

pr

Lesson 2-2: Logic

Law of Syllogism - Example

If it rains today, then we will not see our friends.

Example:

Given:

If it rains today, then we will not have a picnic.

If we do not have a picnic, then we will not see our friends.

Conclusion:

Lesson 2-2: Logic

Inductive Reasoning - Example

- Example 1: Mr. Puyat has worn a SAE t-shirt every Friday for the last couple of weeks. Because of this observation, Franco concludes that Mr. Puyat will wear a SAE t-shirt this Friday.
- What comes next?
- Based on the above, give your own definition of “inductive reasoning”.

Lesson 2-2: Logic

Inductive Reasoning - Definition

- The process of observing data, recognizing patterns, and making conjectures (conclusions) about those patterns

Lesson 2-2: Logic

Deductive Reasoning - Example

- Example 1: Franco saw a school memo saying that all staff must wear their SAE t-shirts on Fridays. Because of this, Franco concludes that Mr. Puyat will wear a SAE t-shirt this Friday.
- Example 2: The area of a whole circle is given by the formula: A = Πr². So if Franco wants to find the shaded area in this diagram , he can use the formula A = ½( Πr²).
- How is deductive reasoning different from inductive reasoning?

Lesson 2-2: Logic

Deductive Reasoning - Definition

- The process of showing that certain statements follow logically from agreed-upon assumptions and proven facts or statements.

Lesson 2-2: Logic

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