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Zorluk: Çok zorOverlapping Sets and Venn Diagrams

At a research institute, 120120 scientists are each assigned to at least one of three interdisciplinary projects: Alpha, Beta, and Gamma. The project assignments meet the following conditions:

- Exactly 7070 scientists work on Project Alpha.
- Exactly 6565 scientists work on Project Gamma.
- Exactly 2525 scientists work on both Project Alpha and Project Beta.
- Exactly 3030 scientists work on both Project Beta and Project Gamma.
- Exactly 2020 scientists work on both Project Alpha and Project Gamma.
- The number of scientists working on all three projects is equal to half the number of scientists who work on Project Gamma only.

How many scientists work on Project Beta only?

  1. 55Cevap
  2. B
    1010
  3. C
    1515
  4. D
    2525
  5. E
    3030

Cevap

Exactly 55 scientists work on Project Beta only.
The correct answer is 55. By using the 3-set Venn diagram equation for Project Gamma, we find the number of scientists in all three projects is x=15x = 15. Decomposing all disjoint regions gives Alpha only =40= 40, Gamma only =30= 30, and two-project overlaps of 1010, 1515, and 55. Subtracting the sum of these six regions (115115) from the total universe (120120) leaves exactly 55 scientists working on Project Beta only.

Adım Adım Çözüm

1
Define variables for the regions of the 3-set Venn diagram.
Let xx be the number of scientists working on all three projects (Alpha ∩\cap Beta ∩\cap Gamma).
Setting the central intersection as xx allows expressing all pairwise overlaps in terms of xx.
2
Express the two-project-only regions and Project Gamma only in terms of xx.
Alpha & Beta only =25−x= 25 - x; Beta & Gamma only =30−x= 30 - x; Alpha & Gamma only =20−x= 20 - x; Gamma only =2x= 2x.
Each given two-set intersection includes the triple intersection xx. Gamma only is twice xx per the stem.
3
Set up an equation for the total number of scientists in Project Gamma (6565) to solve for xx.
Gamma Total=(2x)+(20−x)+(30−x)+x=x+50=65  ⟹  x=15\text{Gamma Total} = (2x) + (20 - x) + (30 - x) + x = x + 50 = 65 \implies x = 15.
Summing all four regions comprising Project Gamma equates to the total given as 6565.
4
Calculate the numerical values of the intersection regions and Gamma only.
All three =15= 15; Gamma only =30= 30; Alpha & Beta only =10= 10; Beta & Gamma only =15= 15; Alpha & Gamma only =5= 5.
Substituting x=15x = 15 back into the expressions from Step 2 gives exact region values.
5
Calculate the number of scientists working on Project Alpha only.
Alpha Total=70=Alpha only+10+5+15  ⟹  Alpha only=40\text{Alpha Total} = 70 = \text{Alpha only} + 10 + 5 + 15 \implies \text{Alpha only} = 40.
Project Alpha is composed of Alpha only plus its three overlapping regions (1010, 55, and 1515).
6
Use the total universe of 120120 scientists to solve for Project Beta only.
120=Alpha only+Beta only+Gamma only+(A∩B)only+(B∩G)only+(A∩G)only+All three  ⟹  120=40+Beta only+30+10+15+5+15=115+Beta only  ⟹  Beta only=5120 = \text{Alpha only} + \text{Beta only} + \text{Gamma only} + (A \cap B)_{\text{only}} + (B \cap G)_{\text{only}} + (A \cap G)_{\text{only}} + \text{All three} \implies 120 = 40 + \text{Beta only} + 30 + 10 + 15 + 5 + 15 = 115 + \text{Beta only} \implies \text{Beta only} = 5.
Since every scientist is on at least one project, the sum of all 7 disjoint region counts equals the total population of 120120.

Anahtar Kavram

Three-Set Overlapping Sets and Venn Diagram Analysis
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Overlapping Sets and Venn Diagrams

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