Set Theory for Competitive Exam Preparation

Set Theory

Q. A group of friends consists of 10 people. If 6 like football, 4 like basketball, and 2 like both, how many like neither sport?

A. 2
B. 4
C. 6
D. 8

Solution

Using the inclusion-exclusion principle: Total liking at least one sport = 6 + 4 - 2 = 8. Therefore, those liking neither = 10 - 8 = 2.

Correct Answer: C — 6

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Q. A group of friends consists of 12 people who like either football or basketball. If 7 like football and 5 like both, how many like only basketball?

A. 5
B. 2
C. 3
D. 4

Solution

Let B be the number of basketball players. Total = Football + Basketball - Both. 12 = 7 + B - 5. Thus, B = 10, and only basketball = 10 - 5 = 5.

Correct Answer: B — 2

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Q. A group of friends consists of 12 people who like either football or basketball. If 7 like football and 5 like basketball, how many like both?

A. 0
B. 2
C. 5
D. 7

Solution

Using the principle of inclusion-exclusion: Total = Football + Basketball - Both. Thus, 12 = 7 + 5 - Both, leading to Both = 0.

Correct Answer: B — 2

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Q. A group of friends consists of 12 people who play football, 8 who play basketball, and 5 who play both. How many play only football?

A. 7
B. 5
C. 8
D. 12

Solution

To find the number of people who play only football, we subtract those who play both from those who play football: 12 - 5 = 7.

Correct Answer: A — 7

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Q. A group of friends consists of 12 who play football, 8 who play basketball, and 5 who play both. How many play only football?

A. 5
B. 8
C. 7
D. 12

Solution

To find the number of friends who play only football, we subtract those who play both from those who play football: 12 - 5 = 7.

Correct Answer: C — 7

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Q. A group of friends consists of 5 people who like football, 3 who like basketball, and 2 who like both. How many like only football? (2023)

A. 3
B. 5
C. 2
D. 0

Solution

To find the number of friends who like only football, we subtract those who like both from those who like football: 5 - 2 = 3.

Correct Answer: A — 3

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Q. A group of friends consists of 5 people who play football, 4 who play basketball, and 2 who play both. How many friends play only one sport?

A. 5
B. 7
C. 9
D. 11

Solution

To find the number of friends who play only one sport, we calculate: (Football only + Basketball only) = (5 - 2) + (4 - 2) = 3 + 2 = 5.

Correct Answer: B — 7

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Q. A group of friends consists of 5 who like football, 4 who like basketball, and 2 who like both. How many friends like only football?

A. 3
B. 4
C. 5
D. 2

Solution

The number of friends who like only football is calculated as: Only Football = Total Football - Both = 5 - 2 = 3.

Correct Answer: A — 3

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Q. If set A = {1, 2, 3, 4} and set B = {3, 4, 5, 6}, what is the difference A - B?

A. {1, 2}
B. {3, 4}
C. {5, 6}
D. {}

Solution

The difference A - B includes elements in A that are not in B, which are {1, 2}.

Correct Answer: A — {1, 2}

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Q. If set A = {1, 2, 3, 4} and set B = {3, 4, 5, 6}, what is the intersection of sets A and B? (2023)

A. {1, 2}
B. {3, 4}
C. {5, 6}
D. {1, 2, 5, 6}

Solution

The intersection of sets A and B includes the elements that are common to both sets. Therefore, the intersection is {3, 4}.

Correct Answer: B — {3, 4}

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Q. If set A = {1, 2, 3} and set B = {2, 3, 4}, what is A - B? (2023)

A. {1}
B. {2, 3}
C. {3, 4}
D. {}

Solution

A - B represents the elements in A that are not in B. Thus, A - B = {1}.

Correct Answer: A — {1}

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Q. If set A = {x | x is an even number less than 10} and set B = {x | x is a prime number less than 10}, what is A ∩ B?

A. {2, 4, 6, 8}
B. {2}
C. {2, 3, 5, 7}
D. {2, 3, 5, 7, 4, 6, 8}

Solution

The intersection A ∩ B includes elements that are both even and prime, which is {2}.

Correct Answer: B — {2}

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Q. If set A contains the elements {1, 2, 3, 4} and set B contains the elements {3, 4, 5, 6}, what is the intersection of sets A and B?

A. {1, 2}
B. {3, 4}
C. {5, 6}
D. {1, 2, 3, 4, 5, 6}

Solution

The intersection of sets A and B is the set of elements that are common to both sets. Therefore, the intersection is {3, 4}.

Correct Answer: B — {3, 4}

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Q. If set A contains the numbers {1, 2, 3, 4, 5} and set B contains the numbers {4, 5, 6, 7, 8}, what is the intersection of sets A and B?

A. {1, 2, 3}
B. {4, 5}
C. {6, 7, 8}
D. {1, 2, 3, 4, 5, 6, 7, 8}

Solution

The intersection of sets A and B is the set of elements that are common to both sets, which is {4, 5}.

Correct Answer: B — {4, 5}

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Q. If set C = {x | x is a multiple of 3 and less than 30}, how many elements are in set C?

A. 8
B. 9
C. 10
D. 11

Solution

The multiples of 3 less than 30 are {3, 6, 9, 12, 15, 18, 21, 24, 27}, totaling 9 elements.

Correct Answer: B — 9

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Q. If set P = {1, 2, 3, 4} and set Q = {3, 4, 5, 6}, what is the difference P - Q?

A. {1, 2}
B. {3, 4}
C. {5, 6}
D. {1, 2, 5, 6}

Solution

The difference P - Q includes elements in P that are not in Q, which is {1, 2}.

Correct Answer: A — {1, 2}

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Q. If set P = {x | x is an even number less than 10} and set Q = {x | x is a prime number less than 10}, what is the union of sets P and Q?

A. {2, 3, 4, 5, 6, 8}
B. {2, 3, 5, 7}
C. {2, 4, 6, 8}
D. {2, 3, 4, 5, 7, 8}

Solution

Set P = {2, 4, 6, 8} and set Q = {2, 3, 5, 7}. The union is {2, 3, 4, 5, 6, 7, 8}.

Correct Answer: D — {2, 3, 4, 5, 7, 8}

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Q. If set P = {x | x is an even number less than 10} and set Q = {x | x is a prime number less than 10}, what is the intersection of sets P and Q?

A. {2, 4, 6, 8}
B. {2, 3, 5, 7}
C. {2}
D. {4, 6, 8}

Solution

The intersection of sets P and Q includes elements that are both even and prime. The only even prime number is 2.

Correct Answer: C — {2}

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Q. If set P = {x | x is an even number less than 10} and set Q = {x | x is a prime number less than 10}, what is the difference P - Q?

A. {2, 4, 6, 8}
B. {4, 6, 8}
C. {2, 6, 8}
D. {2, 4, 6, 8, 3, 5, 7}

Solution

Set P = {2, 4, 6, 8} and set Q = {2, 3, 5, 7}. The difference P - Q = {4, 6, 8}.

Correct Answer: B — {4, 6, 8}

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Q. If set P = {x | x is an even number less than 10} and set Q = {x | x is a prime number less than 10}, what is P ∩ Q?

A. {2, 4, 6, 8}
B. {2, 3, 5, 7}
C. {2}
D. {4, 6, 8}

Solution

The intersection P ∩ Q includes only the even prime number, which is {2}.

Correct Answer: C — {2}

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Q. If set R = {1, 2, 3, 4, 5} and set S = {4, 5, 6, 7}, what is the symmetric difference of sets R and S?

A. {1, 2, 3, 6, 7}
B. {4, 5}
C. {1, 2, 3, 4, 5, 6, 7}
D. {6, 7}

Solution

The symmetric difference of sets R and S includes elements that are in either set but not in both. Thus, it is {1, 2, 3, 6, 7}.

Correct Answer: A — {1, 2, 3, 6, 7}

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Q. If set R = {1, 2, 3, 4} and set S = {3, 4, 5, 6}, what is the difference R - S?

A. {1, 2}
B. {3, 4}
C. {5, 6}
D. {}

Solution

The difference R - S includes elements in R that are not in S, which is {1, 2}.

Correct Answer: A — {1, 2}

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Q. If set R = {1, 2, 3, 4} and set S = {3, 4, 5, 6}, what is the symmetric difference of sets R and S?

A. {1, 2, 5, 6}
B. {3, 4}
C. {1, 2, 3, 4, 5, 6}
D. {3, 4, 5}

Solution

The symmetric difference is the set of elements in either set R or set S but not in both, which is {1, 2, 5, 6}.

Correct Answer: A — {1, 2, 5, 6}

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Q. If set X = {a, b, c} and set Y = {b, c, d}, what is the union of sets X and Y?

A. {a, b, c, d}
B. {b, c}
C. {a, b}
D. {c, d}

Solution

The union of sets X and Y includes all unique elements from both sets. Thus, the union is {a, b, c, d}.

Correct Answer: A — {a, b, c, d}

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Q. If the universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, set A = {2, 4, 6, 8}, and set B = {1, 2, 3}, what is the complement of A?

A. {1, 3, 5, 7, 9, 10}
B. {1, 3, 5, 7, 9}
C. {2, 4, 6, 8}
D. {1, 2, 3}

Solution

The complement of A in U is the set of elements in U that are not in A, which is {1, 3, 5, 7, 9, 10}.

Correct Answer: A — {1, 3, 5, 7, 9, 10}

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Q. If the universal set U = {1, 2, 3, 4, 5, 6} and set A = {2, 4, 6}, what is the complement of set A?

A. {1, 2, 3}
B. {1, 3, 5}
C. {2, 4, 6}
D. {4, 5, 6}

Solution

The complement of set A consists of elements in the universal set U that are not in set A. Therefore, the complement is {1, 3, 5}.

Correct Answer: B — {1, 3, 5}

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Q. If the universal set U has 100 elements, set A has 40 elements, and set B has 30 elements with 10 elements in both A and B, how many elements are in neither A nor B?

A. 60
B. 70
C. 80
D. 50

Solution

Using the principle of inclusion-exclusion, the number of elements in either A or B is: (A + B - Both) = 40 + 30 - 10 = 60. Therefore, elements in neither = U - (A ∪ B) = 100 - 60 = 40.

Correct Answer: A — 60

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Q. In a class of 30 students, 18 students study Mathematics, 15 study Science, and 10 study both subjects. How many students study only Mathematics?

A. 8
B. 10
C. 15
D. 18

Solution

To find the number of students who study only Mathematics, we use the formula: Only Mathematics = Total Mathematics - Both subjects. Thus, 18 - 10 = 8.

Correct Answer: A — 8

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Q. In a class of 40 students, 25 study English, 15 study Mathematics, and 10 study both. How many students study only English?

A. 15
B. 25
C. 10
D. 5

Solution

Students who study only English = Total English - Both = 25 - 10 = 15.

Correct Answer: A — 15

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Q. In a class of 40 students, 25 study English, 20 study Mathematics, and 10 study both. How many study only Mathematics?

A. 10
B. 15
C. 20
D. 25

Solution

To find the number of students who study only Mathematics, we subtract those who study both from those who study Mathematics: 20 - 10 = 10.

Correct Answer: B — 15

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Set Theory

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