Describe each set by referring to the numbered regions, as in Example
X ∩ Y ∩ W
Naming Regions of a Three-Set Venn Diagram
Use figure (a) below to answer each question. question.
a) Name the set we get by combining regions r6 and r7
b) What is a set name for region r2?
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)
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′
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)
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