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Zorluk: OrtaSets, Set Operations, and Venn Diagrams

In a survey of 100100 book club members regarding their reading preferences among Science Fiction (SS), Mystery (MM), and Historical Fiction (HH), it was found that 4848 read Science Fiction, 4242 read Mystery, and 3838 read Historical Fiction. Furthermore, 1818 read both Science Fiction and Mystery, 1515 read both Science Fiction and Historical Fiction, 1414 read both Mystery and Historical Fiction, and 88 read all three genres. How many of the members read exactly one of these three genres?

Cevap: 58 members

Cevap

The number of members who read exactly one of the three genres is 5858.
By isolating the disjoint regions of the 3-set Venn diagram, the number of members reading only Science Fiction is 2323, only Mystery is 1818, and only Historical Fiction is 1717. Adding these disjoint sets yields 5858.

Adım Adım Çözüm

1
Determine exclusive two-set intersection regions
n(S∩M only)=10n(S \cap M \text{ only}) = 10, n(S∩H only)=7n(S \cap H \text{ only}) = 7, n(M∩H only)=6n(M \cap H \text{ only}) = 6
The given pairwise totals include members who read all three genres, so subtracting n(S∩M∩H)=8n(S \cap M \cap H) = 8 isolates those in exactly two sets.
2
Determine exclusive single-set regions
n(S only)=23n(S \text{ only}) = 23, n(M only)=18n(M \text{ only}) = 18, n(H only)=17n(H \text{ only}) = 17
Subtracting all overlapping regions within each set's boundary gives the number of members reading only that specific genre.
3
Sum the single-set regions
23+18+17=5823 + 18 + 17 = 58
The set of members reading exactly one genre is the disjoint union of the three exclusive single-set regions.

Anahtar Kavram

3-Set Venn Diagram Cardinality and Disjoint Region Analysis
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Konular

Sets, Set Operations, and Venn Diagrams

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