Describe each set by referring to the numbered regions, as in ExampleW -(X ∪ Y)Naming Regi...

Describe each set by referring to the numbered regions, as in Example

W -(X ∪ Y)

Naming Regions of a Three-Set Venn Diagram

Use figure (a) below to answer each question. question.

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a)Name the set we get by combining regions r6 and r7

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b) What is a set name for region r2?

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c) Express A - B as the union of two sets.

Solution:

a) Regions r6 and r7 are precisely the regions common to B and C; therefore, this set is B ∩ C. See figure (b)

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b) Region r2 is clearly part of set A; however, elements in region r2 are not elements of B, nor are they elements of C. Therefore, one name for this set is A ∩ B′ ∩ C′

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c) A - B consists of regions r2 and r5, which have set names A ∩ B′ ∩ C′ and A ∩ B′ ∩ C

Notice that another way you can view this set is to start with set A (regions r2, r3, r5, r6) and remove those regions that are in B ∪ C namely r3, r5 and r6. Thus, we also can name region r2 as A - (B ∪ C)

(a) A Venn diagram with three sets. (b) Venn diagram of B ∩ C (c) Venn diagram of A ∩ B ∩ C (d) Venn diagram of A - B.

We can write A – B = (A ∩ B′ ∩ C′) ∪ (A ∩ B′ ∩ C). See figure (d)