GRE Quantitative Quiz: Sets Venn Diagrams

17 questions · exam conditions

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Sets Venn DiagramsQuestion 1 of 17

At a conference of $200$ attendees, $110$ are in set $T$ (attendees who attended a technology session) and $95$ are in set $B$ (attendees who attended a business session). If $40$ attendees attended both types of sessions, what is the number of attendees that attended at least one of the two types of sessions?

205

130

165

150

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GRE Quantitative Quiz

GRE Quantitative Quiz: Sets Venn Diagrams

Practice Sets Venn Diagrams in GRE Quantitative with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Sets Venn Diagrams, giving you a quick way to practice the rules, question types, and explanations that matter most for GRE Quantitative.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

55
205
130
165 (correct answer)
150

Explanation: This question tests the inclusion-exclusion principle for finding the union of two sets. We need to find |T∪B|, the number of attendees who attended at least one session type. Using the formula |T∪B| = |T| + |B| - |T∩B|, we substitute the given values: |T∪B| = 110 + 95 - 40 = 165 attendees. This represents all attendees who attended either a technology session, a business session, or both. A common error would be to simply add 110 + 95 = 205, which double-counts the 40 attendees who attended both types of sessions.

Question 2

In a class of $60$ students, $28$ are in set $S$ (students who play soccer) and $24$ are in set $B$ (students who play basketball). If $10$ students play both soccer and basketball, what is the number of students who play soccer but not basketball?

12
14
38
18 (correct answer)
8

Explanation: This question tests set difference reasoning, specifically finding elements in one set but not another. We need students who play soccer but not basketball, which is |S| - |S∩B|. Given that |S| = 28 and |S∩B| = 10, the number of students who play only soccer is 28 - 10 = 18. This represents the portion of the soccer-playing students who are not in the intersection with basketball players. A common mistake would be to calculate |S| + |B| - |S∩B| = 28 + 24 - 10 = 42, which gives the total in either sport rather than just soccer alone.

Question 3

In a survey of $120$ employees, $70$ are in set $R$ (employees who work remotely at least one day per week) and $55$ are in set $F$ (employees who have flexible hours). If $30$ employees are in both $R$ and $F$ , what is the number of employees that are in exactly one of the sets $R$ or $F$ ?

50
35
95
65 (correct answer)
25

Explanation: This question tests Venn diagram reasoning for finding elements in exactly one set. We have 120 employees total, with |R| = 70, |F| = 55, and |R∩F| = 30. To find employees in exactly one set, we need those in R only plus those in F only. Employees in R only = 70 - 30 = 40, and employees in F only = 55 - 30 = 25. Therefore, employees in exactly one set = 40 + 25 = 65. A common mistake is to calculate |R∪F| = 70 + 55 - 30 = 95, which gives all employees in at least one set, not exactly one.

Question 4

A club has 150 members. Let set $R$ be members who have renewed their membership and set $V$ be members who volunteered this year. If $|R|=98$ , $|V|=64$ , and $|R\cup V|=120$ , what is the number of members who are in both $R$ and $V$ ?

34
22
56
42 (correct answer)
18

Explanation: This question tests the inclusion-exclusion principle when given the union size directly. We need to find |R∩V| using the formula |R∪V| = |R| + |V| - |R∩V|. Substituting the given values: 120 = 98 + 64 - |R∩V|. Solving for the intersection: |R∩V| = 98 + 64 - 120 = 42. This represents the 42 members who both renewed their membership and volunteered. A common mistake would be to subtract the given values incorrectly or to confuse the union with the intersection when setting up the equation.

Question 5

At a company, let set $R$ be employees who work remotely and set $T$ be employees who have completed technical training. There are 80 employees total. If $|R|=46$ , $|T|=38$ , and $|R\cap T|=20$ , what is the number of employees who are in exactly one of the two sets $R$ and $T$ ?

64
44 (correct answer)
26
24
84

Explanation: This question tests Venn diagram reasoning to find elements in exactly one set. We need to find employees who are in R but not T, plus those in T but not R. First, calculate |R only| = |R| - |R ∩ T| = 46 - 20 = 26 employees who work remotely but haven't completed training. Next, calculate |T only| = |T| - |R ∩ T| = 38 - 20 = 18 employees who completed training but don't work remotely. The total in exactly one set is 26 + 18 = 44. A common mistake would be to calculate |R ∪ T| = 64 instead, which includes those in both sets.

Question 6

A conference has 110 attendees. Let set $A$ be attendees who attended Workshop 1 and set $B$ be attendees who attended Workshop 2. If $|A|=63$ , $|B|=57$ , and 22 attendees attended neither workshop, what is the number of attendees who attended both workshops?

40
10
88
32 (correct answer)
14

Explanation: This question tests the inclusion-exclusion principle to find the intersection of two sets. We know 22 attended neither workshop, so |A∪B| = 110 - 22 = 88 attendees attended at least one workshop. Using inclusion-exclusion: |A∪B| = |A| + |B| - |A∩B|, we get 88 = 63 + 57 - |A∩B|. Solving for the intersection: |A∩B| = 63 + 57 - 88 = 32. This represents the 32 attendees who attended both workshops. A common mistake would be to add those who attended neither to the union calculation, giving an incorrect result.

Question 7

In a town of 150 households, let set $C$ be households that own a cat and set $O$ be households that own a dog. If 40 households own neither a cat nor a dog, and $|C\cap O|=25$ , and $|C|=70$ , what is the number of households that own a dog?

65 (correct answer)
40
105
45
95

Explanation: This question tests finding a set's cardinality given partial information. We know 150 total households, 40 own neither pet, so 110 own at least one pet. We have |C| = 70 and |C ∩ O| = 25. Using |C ∪ O| = 110 and the inclusion-exclusion formula: 110 = 70 + |O| - 25. Solving for |O|: |O| = 110 - 70 + 25 = 65 households own a dog. A common error would be to calculate |O| = 110 - 70 = 40, forgetting to add back the intersection that was subtracted when finding the union.

Question 8

In a group of 80 employees, let set $A$ be those who work remotely at least 1 day per week and set $B$ be those who use public transportation to commute. If $|A|=46$ , $|B|=38$ , and $|A\cap B|=19$ , what is the number of employees who are in neither $A$ nor $B$ ?

26
34
23
15 (correct answer)
42

Explanation: This question tests set reasoning using the principle of inclusion-exclusion for two sets. We need to find the number of employees in neither set A (remote workers) nor set B (public transport users), which equals the total minus those in A∪B. Using the inclusion-exclusion principle: |A∪B| = |A| + |B| - |A∩B| = 46 + 38 - 19 = 65. Therefore, the number in neither set is 80 - 65 = 15. A common error would be to add |A| and |B| without subtracting the intersection, giving 84 employees in the union, which would incorrectly suggest -4 employees are in neither set.

Question 9

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 $|M\cap P|=15$ , what is the number of students who study neither Math nor Physics?

12 (correct answer)
32
18
25
8

Explanation: This question tests set reasoning using the principle of inclusion-exclusion. We can model this with a Venn diagram where the universal set contains all 60 students, with two overlapping circles representing sets M (Math students) and P (Physics students). To find students studying at least one subject, we use |M ∪ P| = |M| + |P| - |M ∩ P| = 35 + 28 - 15 = 48. Therefore, the number of students studying neither subject is 60 - 48 = 12. A common error would be to add 35 + 28 = 63 without subtracting the overlap, which would incorrectly count the 15 students in both sets twice.

Question 10

In a class of 80 students, let set $F$ be those who speak French and set $S$ be those who speak Spanish. If $|F|=42$ , $|S|=38$ , and $|F\cup S|=65$ , what is the number of students who speak both French and Spanish?

15 (correct answer)
7
80
27
20

Explanation: This question tests set theory reasoning using Venn diagrams for two overlapping sets. We model the students speaking French (F) and Spanish (S) with regions for only F, only S, and both. The union |F ∪ S| = 65 is given, so the intersection is 42 + 38 - 65 = 15. This allocates only French as 42 - 15 = 27 and only Spanish as 38 - 15 = 23. The result is justified by the inclusion-exclusion principle avoiding double-counting. A common incorrect option is 20, perhaps from subtracting the union from one set alone, like 42 + 38 - 80 = 0, but this ignores the given union and leads to errors.

Question 11

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?

15
25 (correct answer)
70
5
30

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.

Question 12

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
45
33
35 (correct answer)
27

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.

Question 13

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
45 (correct answer)
55
25
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.

Question 14

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?

12 (correct answer)
3
10
32
18

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.

Question 15

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?

75
55 (correct answer)
35
25
60

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.

Question 16

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 (correct answer)
36
26
28
10

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.

Question 17

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?

140 (correct answer)
230
70
90
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.