1. Lesson 2-2: Logic 1 Lesson 2-2 Logic

2. 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. B DOGS.A.A...B   dog

3. 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

4. Lesson 2-2: Logic 4 To Show Relationships using Venn Diagrams: Blue or Brown (includes Purple) …A  B AB A  B

5. 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...

6. Lesson 2-2: Logic 6 Solution: Three members were on cross country and football teams… 3 The above sentence tells you two draw overlapping circles and put 3 in CC  F 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. CCFootball 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….

7. 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 3 5 1 Band 15 Mu Alpha Theta

8. Lesson 2-2: Logic 8 Law of Detachment Given: a true conditional statement and the hypothesis occurs p  q is true p is given Conclusion: the conclusion will also occur q is true

9. 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:

10. Lesson 2-2: Logic 10 Law of Syllogism Given: Two true conditional statements and the conclusion of the first is the hypothesis of the second. p  q and q  r Conclusion: If the hypothesis of the first occurs, then the conclusion of the second will also occur. prpr

11. 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:

12. Lesson 2-2: Logic 12 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”. Inductive Reasoning - Example

13. Lesson 2-2: Logic 13 The process of observing data, recognizing patterns, and making conjectures (conclusions) about those patterns Inductive Reasoning - Definition

14. Lesson 2-2: Logic 14 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? Deductive Reasoning - Example

15. Lesson 2-2: Logic 15 The process of showing that certain statements follow logically from agreed- upon assumptions and proven facts or statements. Deductive Reasoning - Definition