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Understanding Logic: Conditional Statements and Venn Diagrams

In this lesson, we explore fundamental concepts of logic through conditional statements and Venn diagrams. We illustrate how Venn diagrams can represent relationships between different data sets and conditional logic. By applying the Law of Detachment and the Law of Syllogism, we solve real-world problems, like determining the membership distribution in clubs. With engaging examples, including congruent angles and conditional relationships, this lesson aims to deepen your understanding of logical reasoning and its applications.

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Understanding Logic: Conditional Statements and Venn Diagrams

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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

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