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Lesson 2-2

Logic. Lesson 2-2. DOGS. 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.

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Lesson 2-2

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

  2. DOGS 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

  3. 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

  4. To Show Relationships using Venn Diagrams: Blue or Brown (includes Purple) … AB 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: 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 CCF • 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

  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 Band 3 5 1 15 Mu Alpha Theta 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 Lesson 2-2: Logic

  9. 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

  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r Lesson 2-2: Logic

  11. 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

  12. 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

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

  14. 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

  15. 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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