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Venn Diagrams and Logic

Lesson 2-2. Venn Diagrams and Logic. Venn diagrams :. show relationships between different sets of data. can represent conditional statements. DOGS. ...B   dog. A=poodle ... a dog. . A. B= horse ... NOT a dog. B. A Venn diagram is usually drawn as a circle.

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Venn Diagrams and Logic

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  1. Lesson 2-2 Venn Diagrams andLogic

  2. Venn diagrams: • show relationships between different sets of data. • can represent conditional statements.

  3. DOGS ...B  dog A=poodle ... a dog .A B= horse ... NOT a dog . B A Venn diagram is usually drawn as a circle. • Every point IN the circlebelongs to that set. • Every point OUT of the circledoes not.

  4. Many Venn diagrams are drawn as two overlapping circles. A is in A is not in Group 1 Group 2

  5. Many Venn diagrams are drawn as two overlapping circles B is in Group 1 AND Group 2

  6. Many Venn diagrams are drawn as two overlapping circles C is in C is not in Group 2 Group 1

  7. Many Venn diagrams are drawn as two overlapping circles Some of the elements in group1 are in group2 Some of the elements in group2 are in group1

  8. Sometimes the circles do not overlap in a Venn diagram. D is in D is not in Group 3 Group 4

  9. Sometimes the circles do not overlap in a Venn diagram. E is in E is not in Group 4 Group 3

  10. Sometimes the circles do not overlap in a Venn diagram. None of the elements in group3 are in group4 None of the elements in group4 are in group3

  11. In Venn diagrams it is possible to have one circle inside another. F is in Group 5 AND is in Group 6

  12. In Venn diagrams it is possible to have one circle inside another. G is in G is not in Group 6 Group 5

  13. In Venn diagrams it is possible to have one circle inside another. All of the elements in group5 are in group6 Some of the elements in group6 are in group5

  14. congruent angles • right angles If two angles are right angles, then they are congruent. All right angles are congruent.

  15. flower If you have a rose, then you have a flower. Every rose is a flower. rose

  16. Lines that do not • intersect If two lines are parallel, then they do not intersect. parallel lines

  17. Let’s see how this works!Suppose you are given ... 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?

  18. Use a Venn Diagram and take one sentence at a time... • Three members were on cross country and football teams… • Tells you two draw overlapping circles • Put 3 marks in CCF

  19. Use a Venn Diagram and take one sentence at a time... • One-third of the members ran cross country. • put 8 marks in the CC circle since there are 24 members • already 3 marks • so put 5 marks in the red part

  20. Use a Venn Diagram and take one sentence at a time... • One sixth of the members were on the football team . • put 4 marks in the Football circle since there are 24 members • already 3 marks • so put 1 mark in the purple part IIIII

  21. Use a Venn Diagram and take one sentence at a time... • 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 • 15 members are in band

  22. Drawing and Supporting Conclusions

  23. Law of Detachment You are given: a true conditional statement and the hypothesis occurs You can conclude: that the conclusion will also occur

  24. Law of Detachment You are given: pq is true p is given You can conclude: q is true Symbolic form

  25. Law of Detachment Example You are given: If three points are collinear, then the points are all on one line. E,F, and G are collinear. You can conclude: E,F, and G are all on one line.

  26. Law of Syllogism You are given: Two true conditional statements and the conclusion of the first is the hypothesis of the second. You can conclude: that if the hypothesis of the first occurs, then the conclusion of the second will also occur

  27. Law of Syllogism You are given: pq and qr You can conclude: pr Symbolic form

  28. Law of Syllogism Example You are 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. You can conclude: If it rains today, then we will not see our friends.

  29. Series of Conditionals The law of syllogism can be applied to a series of statements. Simply reorder statements.

  30. You may need to use contrapositives, since they are logical equivalents to the original statement. This means 1) ~st is the same as 2) s  r is the same as 3) r  ~q is the same as ~t  s ~r  ~s q  ~r

  31. EXAMPLE: if p  q, s  r, ~s  t, and r  ~q; then p  ______? You might need the contrapositives: ~t  sor~r  ~sor q  ~r Start with p and use the law of syllogism to find the conclusion: p  q … ~s  t … q  ~r … ~r  ~s  p  t

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