Venn Diagrams and Logic

Venn Diagrams andLogicLesson 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.

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 circlesB is inGroup 1 AND Group 2

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

Many Venn diagrams are drawn as two overlapping circlesof the elements in group1 are in group2Someof the elements in group2 are in group1Some

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 group4Noneof the elements in group4 are in group3None

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 group6Allof the elements in group6 are in group5Some

All right angles are congruent.

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

Every rose is a flower.rose

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

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

Lets 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 teamsTells you two draw overlapping circlesPut 3 marks in CCF

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 marksin 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 markin the purple partIIIII

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 country15 membersare in band

Drawing and Supporting Conclusions

Law of DetachmentYou are given:a true conditional statement andthe hypothesis occurs

You can conclude:that the conclusion will also occur

Law of DetachmentYou are given:pq is truep is given

You can conclude:q is true

Symbolic form

Law of DetachmentYou 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 SyllogismYou are given:Two true conditional statements andthe 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 SyllogismYou are given:pq and qr

You can conclude:prSymbolic form

Law of SyllogismYou 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 ConditionalsThe 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 means1) ~st is the same as2) s r is the same as3) r ~q is the same as

~t s~r ~sq ~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 ~rStart with p and use the law of syllogism to find the conclusion:p q q ~r ~r ~s ~s t p t