Venn Diagrams and Logic Lesson 2-2. Venn diagrams: show relationships between different sets of data. can represent conditional statements.

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Presentation transcript:

Venn Diagrams and Logic Lesson 2-2

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

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

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

Many Venn diagrams are drawn as two overlapping circles B is inGroup 1 AND Group 2

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

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

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

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

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

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

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

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

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

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

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

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?

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

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

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

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

Drawing and Supporting Conclusions

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

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

Law of Detachment 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. Example

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

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

Law of Syllogism 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. Example

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

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

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