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Solving A Logic Problem with A Venn Diagram

Created by E.G. Gascon

Problem Section 7.2 #41

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The problem

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State the Problem in Set Notation

n(T) = 22n(G) = 25n(S) = 39n(TG) = 9n(GS) = 20n(T’G’S’) = 4n(TGS) = 6

I switched the last two because the 6 is the intersections of all the circles in the Venn Diagram.

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Building the Venn Diagram from the inside out.

Start in the inner most intersection of the diagram, where the three circles all intersect. n(TGS) = 6

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Label the area that has NO items

n(T’G’S’) = 4

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Build out from the center.

n(GS) = 20

Next enter the number for the areas where only two circles overlap.

n(TG) = 9 But, …Part of the intersection of T and G is already = 6

But, … Part of the intersection of G and S is already = 6

What is left, will be placed in the next areas.

Notice that n(TS) does not have a value. Let it be x.

x

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Continue to work up the list

n(S) = 39 All of S = 39, but there are already 6 + 14 + x object in the set S.

Therefore, what is exclusively S is

39 – 6 – 14 - x

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Continue to work up the list

n(G) = 25 All of S = 25, but there are already 6 + 14 + 3 object in the set G.

Therefore, what is exclusively G is

25 – 6 – 14 - 3

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Continue to work up the list

n(T) = 22 All of T = 22, but there are already 6 + 3 + x object in the set T.

Therefore, what is exclusively T is

22 – 6 – 3 – x

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Create an equation from the information:

4 13 19 2 14 3 6 50

61 50

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11

13 2

19 8

x x x

x

x

x

x

x

Add all the components of the sample space:

You are now ready to answer the questions.

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a) Tall and Smooth = 11 + 6 = 17

Look at the circle that represents the Tall and the circle that represents the Smooth. Now consider where they overlap.

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b.) Tall and NOT (smooth nor Green) = 2

Look at the circle that represent Tall, now take away the whatever is smooth or green in the circle.

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c) Not Tall but had peas that were smooth and green = 14

Look outside of the circle that represents Tall. Then consider only the area of overlap between Smooth and Green.

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Questions?????

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