Sets, Venn Diagrams, and Overlap
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GRE Quantitative Reasoning › Sets, Venn Diagrams, and Overlap
In a class of 90 students, let set $G$ be those who play a musical instrument and set $H$ be those who play a sport. If $|G|=40$, $|H|=55$, and $|G\cup H|=70$, what is the number of students who play both a musical instrument and a sport?
5
15
25
30
70
Explanation
This question tests set theory reasoning using Venn diagrams for two overlapping sets. We model the students playing instrument (G) and sport (H) with regions for only G, only H, and both. The union |G ∪ H| = 70 is given, so the intersection is 40 + 55 - 70 = 25. This allocates only G as 40 - 25 = 15 and only H as 55 - 25 = 30. The result is justified by inclusion-exclusion avoiding double-counting. A common incorrect option is 15, perhaps from subtracting one set from union, like 70 - 55 = 15, ignoring the full overlap calculation.
In a group of 160 students, let set $A$ be those enrolled in Art, set $B$ be those enrolled in Biology, and set $C$ be those enrolled in Chemistry. If $|A|=60$, $|B|=70$, $|C|=55$, $|A\cap B|=25$, $|A\cap C|=20$, $|B\cap C|=18$, and $|A\cap B\cap C|=8$, what is the number of students enrolled in Biology only (in $B$ but not in $A$ or $C$)?
19
27
33
35
45
Explanation
This question tests set theory reasoning using Venn diagrams for three overlapping sets. We model the students in art (A), biology (B), and chemistry (C) with regions for only one, exactly two, and all three. The triple intersection is 8; exactly A and B is 25 - 8 = 17, A and C is 20 - 8 = 12, B and C is 18 - 8 = 10. Only B is 70 - 25 - 18 + 8 = 35. This is justified by subtracting pairwises and adding back triple for the exclusive B region. A common incorrect option is 27, perhaps from forgetting to add back the triple, like 70 - 25 - 18 = 27, undercounting the adjustment for overlap.
In a group of 140 applicants, let set $E$ be those with prior work experience and set $D$ be those with a relevant degree. If 30 applicants have neither, $|E|=85$, and $|D|=70$, what is the number of applicants who have both prior work experience and a relevant degree?
15
25
45
55
115
Explanation
This question tests set theory reasoning using Venn diagrams for two overlapping sets. We model the applicants with experience (E) and degree (D) with regions for only E, only D, and both. With 30 having neither, the union is 140 - 30 = 110. The intersection is 85 + 70 - 110 = 45. This is justified by rearranging the inclusion-exclusion formula. A common incorrect option is 15, perhaps from subtracting neither from one set, like 85 - 30 - 70 or other misallocations, undercounting the overlap.
In a group of 60 students, let set $M$ be the students who study Math and set $P$ be the students who study Physics. If $|M|=35$, $|P|=28$, and $|M\cap P|=15$, how many students study neither Math nor Physics?
3
10
12
18
32
Explanation
This question tests set theory reasoning using Venn diagrams for two overlapping sets. We model the students in Math (M) and Physics (P) with regions for only M, only P, and both. The intersection |M ∩ P| = 15 represents students studying both subjects. The number studying only Math is 35 - 15 = 20, and only Physics is 28 - 15 = 13. The total studying at least one subject is 20 + 13 + 15 = 48, so those studying neither is 60 - 48 = 12. A common incorrect option arises from adding |M| and |P| without subtracting the intersection, leading to 35 + 28 = 63 and 60 - 63 = -3, which is impossible and highlights the error of double-counting the overlap.
In a group of $90$ people, $50$ are in set $G$ (people who have been to Germany) and $45$ are in set $I$ (people who have been to Italy). If $20$ people have been to both Germany and Italy, what is the number of people who have been to exactly one of the two countries?
25
35
55
60
75
Explanation
This question tests finding elements in exactly one of two sets. We need people who visited exactly one country, which equals those who visited Germany only plus those who visited Italy only. People who visited only Germany = |G| - |G∩I| = 50 - 20 = 30, and people who visited only Italy = |I| - |G∩I| = 45 - 20 = 25. Therefore, people who visited exactly one country = 30 + 25 = 55. A common mistake would be to calculate |G∪I| - |G∩I| = (50 + 45 - 20) - 20 = 55, which happens to give the same answer but uses incorrect reasoning.
In a survey of 100 employees, let set $T$ be those who took a training course and set $C$ be those who earned a certification. If $|T|=64$, $|C|=46$, and 18 employees did both, what is the number of employees who did neither?
8
10
26
28
36
Explanation
This question tests set theory reasoning using Venn diagrams for two overlapping sets. We model the employees taking training (T) and earning certification (C) with regions for only T, only C, and both. The intersection |T ∩ C| = 18 is given, so the union is 64 + 46 - 18 = 92. The number who did neither is 100 - 92 = 8. This is justified by the inclusion-exclusion for union. A common incorrect option is 10, perhaps from adding the intersection instead of subtracting, like 64 + 46 + 18 - 100 or miscalculations leading to overestimation.
In a town of 500 residents, let set $H$ be those who have a home garden and set $P$ be those who participate in a composting program. If $|H|=210$, $|P|=160$, and $|H\cap P|=70$, what is the number of residents who have a home garden but do not participate in the composting program?
70
90
140
230
280
Explanation
This question tests set theory reasoning using Venn diagrams for two overlapping sets. We model the residents with home garden (H) and composting (P) with regions for only H, only P, and both. The intersection |H ∩ P| = 70 is given, so only H is 210 - 70 = 140. This directly allocates the exclusive region for H. The result is justified as the set difference |H| - |H ∩ P|. A common incorrect option is 230, perhaps from adding sets without subtracting intersection, like 210 + 160 - 140 or errors in union calculation.