Constraint-Based Sets for Competitive Exam Success

Constraint-Based Sets

Q. Consider the set defined by the constraints: {x | x is an even number and x > 0}. Which of the following is a valid member of this set?

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

Solution

The number 2 is the only even number greater than 0 among the options, thus it is a valid member of the set.

Correct Answer: B — 2

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Q. If a constraint-based set is defined as 'the set of all x such that x is a multiple of 3 and x < 30', which of the following is an element of this set?

A. 27
B. 29
C. 31
D. 15

Solution

27 is a multiple of 3 and is less than 30, making it an element of the set.

Correct Answer: A — 27

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Q. If a constraint-based set is defined as 'the set of all x such that x is a natural number and x < 10', which of the following is a valid representation of this set?

A. {1, 2, 3, 4, 5, 6, 7, 8, 9}
B. {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
C. {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
D. {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}

Solution

The correct representation of the set is {1, 2, 3, 4, 5, 6, 7, 8, 9}, as it includes all natural numbers less than 10.

Correct Answer: A — {1, 2, 3, 4, 5, 6, 7, 8, 9}

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Q. If a constraint-based set is defined as 'the set of all x such that x is an integer and 1 ≤ x ≤ 5', which of the following is a valid representation of this set?

A. {1, 2, 3, 4, 5}
B. {0, 1, 2, 3, 4, 5}
C. {1, 2, 3, 4, 5, 6}
D. {2, 3, 4, 5, 6}

Solution

The valid representation of the set is {1, 2, 3, 4, 5} as it includes all integers within the specified range.

Correct Answer: A — {1, 2, 3, 4, 5}

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Q. If a constraint-based set is defined as {x | x is a letter in the English alphabet and x is a vowel}, which of the following letters is included in this set?

A. B
B. C
C. A
D. D

Solution

The letter 'A' is included in the set of vowels in the English alphabet.

Correct Answer: C — A

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Q. If a constraint-based set is defined as {x | x is a multiple of 3 and x < 30}, how many elements does this set contain?

A. 9
B. 10
C. 8
D. 7

Solution

The multiples of 3 less than 30 are 3, 6, 9, 12, 15, 18, 21, 24, 27, totaling 10 elements.

Correct Answer: B — 10

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Q. If a constraint-based set is defined as {x | x is a multiple of 3 and x < 30}, which of the following is NOT an element of this set?

A. 3
B. 9
C. 27
D. 31

Solution

The number 31 is not an element of the set as it does not meet the constraints of being a multiple of 3 and less than 30.

Correct Answer: D — 31

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Q. If a constraint-based set is defined as {x | x is a real number and x^2 < 4}, what is the range of x?

A. (-2, 2)
B. [0, 2]
C. (-∞, ∞)
D. (0, 4)

Solution

The inequality x^2 < 4 implies that x must be between -2 and 2, hence the range is (-2, 2).

Correct Answer: A — (-2, 2)

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Q. If a set A contains elements {1, 2, 3, 4, 5} and a constraint states that only even numbers can be included, what will be the resulting set?

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

Solution

The resulting set will be {2, 4} as these are the only even numbers in set A.

Correct Answer: A — {2, 4}

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Q. If a set A is defined by the constraints x > 5 and x < 10, which of the following numbers is NOT a member of set A?

A. 6
B. 7
C. 10
D. 9

Solution

The number 10 does not satisfy the constraint x < 10, hence it is not a member of set A.

Correct Answer: C — 10

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Q. If a set D is defined as {x | x is a positive integer and x is less than or equal to 15}, how many elements are in set D?

A. 10
B. 15
C. 14
D. 16

Solution

Set D includes all positive integers from 1 to 15, which totals 15 elements.

Correct Answer: B — 15

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Q. If a set is defined as containing all prime numbers less than 20, which of the following numbers is NOT in the set?

A. 2
B. 3
C. 4
D. 19

Solution

The number 4 is not a prime number, hence it is not included in the set of prime numbers less than 20.

Correct Answer: C — 4

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Q. If a set is defined as {x | x is a fruit and x is red}, which of the following fruits would NOT be included in this set?

A. Apple
B. Banana
C. Cherry
D. Strawberry

Solution

The fruit 'Banana' would not be included in the set as it is not red.

Correct Answer: B — Banana

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Q. If a set is defined as {x | x is a letter in the English alphabet and x is a consonant}, which of the following letters is NOT included in this set?

A. B
B. E
C. K
D. T

Solution

The letter E is a vowel and therefore not included in the set of consonants.

Correct Answer: B — E

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Q. If a set is defined as {x | x is a letter in the English alphabet and x is a vowel}, which of the following letters is NOT an element of this set?

A. A
B. E
C. I
D. Z

Solution

The letter Z is not a vowel and therefore is not an element of the set defined by vowels.

Correct Answer: D — Z

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Q. If a set is defined as {x | x is a letter in the English alphabet and x is a vowel}, which of the following is an element of this set?

A. B
B. C
C. A
D. D

Solution

The letter A is an element of the set as it is a vowel in the English alphabet.

Correct Answer: C — A

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Q. If a set is defined as {x | x is a letter in the English alphabet that is a vowel}, which of the following letters is NOT in the set?

A. A
B. E
C. I
D. Y

Solution

The vowels in the English alphabet are A, E, I, O, and U. The letter Y is sometimes considered a vowel but is not classified as one in this context.

Correct Answer: D — Y

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Q. If a set is defined as {x | x is a natural number and x < 10}, which of the following is the correct representation of this set?

A. {1, 2, 3, 4, 5, 6, 7, 8, 9}
B. {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
C. {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
D. {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}

Solution

The correct representation of the set is {1, 2, 3, 4, 5, 6, 7, 8, 9}, as it includes all natural numbers less than 10.

Correct Answer: A — {1, 2, 3, 4, 5, 6, 7, 8, 9}

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Q. If a set is defined as {x | x is a natural number and x < 10}, which of the following is the largest element in this set?

A. 8
B. 9
C. 10
D. 7

Solution

The largest natural number less than 10 is 9, making it the largest element in the defined set.

Correct Answer: B — 9

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Q. If a set is defined as {x | x is a natural number and x < 10}, which of the following is a valid element of this set?

A. 10
B. 5
C. 0
D. 15

Solution

The valid element of the set is 5, as it is a natural number and less than 10.

Correct Answer: B — 5

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Q. If a set is defined as {x | x is a natural number and x < 5}, which of the following elements is NOT part of this set?

A. 1
B. 3
C. 5
D. 4

Solution

The element '5' is not part of the set because the constraint specifies that x must be less than 5.

Correct Answer: C — 5

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Q. If a set is defined as {x | x is a natural number less than 20 and even}, which of the following numbers is included in the set?

A. 1
B. 3
C. 10
D. 21

Solution

The even natural numbers less than 20 are 2, 4, 6, 8, 10, 12, 14, 16, and 18. Therefore, 10 is included in the set.

Correct Answer: C — 10

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Q. If a set is defined as {x | x is a positive integer and x is a divisor of 12}, which of the following is NOT an element of this set?

A. 1
B. 2
C. 3
D. 5

Solution

The divisors of 12 are 1, 2, 3, 4, and 6, so 5 is not an element of this set.

Correct Answer: D — 5

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Q. If a set is defined as {x | x is a positive integer and x is a multiple of 3}, which of the following is NOT an element of this set?

A. 3
B. 6
C. 9
D. 10

Solution

The number '10' is not a multiple of 3, hence it is not an element of the set.

Correct Answer: D — 10

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Q. If a set is defined as {x | x is a prime number less than 10}, which of the following elements is NOT in the set?

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

Solution

The set includes prime numbers less than 10, which are 2, 3, 5, and 7. The number 4 is not a prime number.

Correct Answer: D — 4

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Q. If a set is defined as {x | x is a prime number less than 20}, which of the following elements is NOT included in this set?

A. 2
B. 3
C. 4
D. 19

Solution

The number 4 is not a prime number, hence it is not included in the set of prime numbers less than 20.

Correct Answer: C — 4

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Q. If a set is defined as {x | x is a prime number less than 20}, which of the following elements is NOT included in the set?

A. 2
B. 3
C. 4
D. 19

Solution

The number 4 is not a prime number, hence it is not included in the set of prime numbers less than 20.

Correct Answer: C — 4

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Q. If a set is defined as {x | x is an even number greater than 0 and less than 20}, which of the following is NOT an element of this set?

A. 2
B. 4
C. 10
D. 21

Solution

The number 21 is not included in the set as it does not meet the criteria of being less than 20.

Correct Answer: D — 21

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Q. If a set is defined as {x | x is an integer and 1 ≤ x ≤ 10}, which of the following is a valid operation on this set?

A. Adding 5 to each element.
B. Removing all even numbers.
C. Doubling each element.
D. All of the above.

Solution

All the operations listed can be validly performed on the set defined by the constraints.

Correct Answer: D — All of the above.

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Q. If a set is defined by the constraint of being a natural number, which of the following is included?

A. -1
B. 0
C. 1
D. 2.5

Solution

1 is a natural number, while -1 and 0 are not considered natural numbers, and 2.5 is not an integer.

Correct Answer: C — 1

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Constraint-Based Sets

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