Lesson 2-2: Logic 1

Lesson 2-2

Logic

Lesson 2-2: Logic 2

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: A =poodle ... a dog

B= horse ... NOT a dog

.BDOGS

.A...B dog

Lesson 2-2: Logic 3

For all..., every..., if...then...All right angles are congruent.

Congruent Angles

Right Angles

Example1:

Example 2: Every rose is a flower.

Flower

Rose

Example 3: If two lines are parallel, then they do not intersect.

lines that do not intersect

parallel lines

Lesson 2-2: Logic 4

To Show Relationships using Venn Diagrams:

Blue or Brown (includes Purple) … AB

A B

A B

Lesson 2-2: Logic 5

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 6

Solution:

Three members were on cross country and football teams…

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

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

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.

CC Football

5

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.

1

Continued….

Lesson 2-2: Logic 7

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

35 1

Band

15

Mu Alpha Theta

Lesson 2-2: Logic 8

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 9

Law of Detachment - Example

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 1:

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.

Example 2:

Lesson 2-2: Logic 10

Law of Syllogism

Given:Two true conditional statements andthe 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 11

Law of Syllogism - Example

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

Example:

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:

Given: