Fundamentals Of Set Theory And Relations Assessment Practice Tests

Question 1

Given the set $A = \{x \in \mathbb{Z} \mid x^2 < 10\}$, which of the following represents the set in roster notation?

Accepted Answer

$\{-3, -2, -1, 0, 1, 2, 3\}$

Answer

$\{1, 2, 3\}$

Answer

$\{-4, -3, -2, -1, 0, 1, 2, 3, 4\}$

Answer

$\{0, 1, 2, 3\}$

Question 2

Which of the following collections is NOT a well-defined set?

Accepted Answer

The set of all beautiful paintings in the Louvre.

Answer

The set of all prime numbers less than 20.

Answer

The set of all letters in the word 'MATHEMATICS'.

Answer

The set of all solutions to the equation $x^2 - 4 = 0$.

Question 3

Represent the set $B = \{x \mid x 	ext{ is a positive even integer less than 12}\}$ using roster notation.

Accepted Answer

$\{2, 4, 6, 8, 10\}$

Question 4

Convert the roster set $\{a, e, i, o, u\}$ into set-builder notation.

Accepted Answer

$\{x \mid x 	ext{ is a vowel in the English alphabet}\}$

Answer

$\{x \mid x 	ext{ is a letter in the alphabet}\}$

Answer

$\{x \mid x 	ext{ is a consonant in the English alphabet}\}$

Answer

$\{x \mid x \in \mathbb{Z}, 1 < x < 5\}$

Question 5

How many distinct elements are in the set $S$ formed by the letters of the word 'PROBABILITY'?

Accepted Answer

8

Answer

7

Answer

9

Answer

11

Question 6

Determine the cardinality of the set $C = \{x \in \mathbb{N} \mid 5 < x \leq 10\}$.

Accepted Answer

5

Answer

4

Answer

6

Answer

10

Question 7

The set $D = \{x \in \mathbb{R} \mid x^2 + 1 = 0\}$ is a unit set.

Accepted Answer

False

Answer

True

Question 8

Which of the following sets is classified as an infinite set?

Accepted Answer

$\{x \in \mathbb{Z} \mid x > 0\}$

Answer

$\{x \in \mathbb{Z} \mid -5 < x < 5\}$

Answer

$\{x \in \mathbb{N} \mid x < 100\}$

Answer

$\{x \mid x 	ext{ is a month of the year}\}$

Question 9

Calculate the cardinality of the set $E = \{x \mid x 	ext{ is a factor of } 12\}$.

Accepted Answer

6

Question 10

Identify all sets that are considered finite.

Answer

The set of all days in a week

Answer

The set of all integers

Answer

The set of all prime numbers less than 50

Answer

The set of all real numbers

Question 11

Given sets $A = \{x \in \mathbb{N} : x^2 < 10\}$ and $B = \{1, 2, 3\}$, determine the relationship between $A$ and $B$.

Accepted Answer

A. $A$ and $B$ are equal ($A = B$)

Answer

B. $A$ and $B$ are equivalent but not equal ($A \sim B$)

Answer

C. $A$ is a proper subset of $B$ ($A \subset B$)

Answer

D. Neither equal nor equivalent

Question 12

Consider $A = \{	ext{prime numbers } < 10\}$ and $B = \{2, 3, 5, 7\}$. Which statement is true?

Accepted Answer

B. $A$ and $B$ are both equal and equivalent

Answer

A. $A$ and $B$ are equivalent but not equal

Answer

C. $A$ is a proper subset of $B$

Answer

D. $A$ and $B$ are neither equal nor equivalent

Question 13

Let $A = \{a, b, c\}$ and $B = \{1, 2, 3, 4\}$. Determine the relationship between $A$ and $B$.

Accepted Answer

C. $A$ and $B$ are neither equal nor equivalent

Answer

A. $A$ is equivalent to $B$

Answer

B. $A$ is equal to $B$

Answer

D. $A$ is a proper subset of $B$

Question 14

If $A = \{x \in \mathbb{Z} : -2 < x < 2\}$ and $B = \{-1, 0, 1\}$, then $A \subset B$ is a true statement.

Accepted Answer

True

Answer

False

Question 15

For any set $A$, the empty set $\emptyset$ is always a proper subset of $A$, provided $A 
eq \emptyset$.

Accepted Answer

True

Answer

False

Question 16

If $A \subset B$, then it is impossible for $A$ to be equal to $B$.

Accepted Answer

True

Answer

False

Question 17

If $A = \{1, 2\}$, then the set $\{1, 2\}$ is a proper subset of $A$.

Accepted Answer

False

Answer

True

Question 18

Calculate the cardinality of the power set $P(A)$ where $A = \{a, b, c, d, e\}$.

Accepted Answer

32

Question 19

If the cardinality of a power set $|P(A)|$ is $64$, determine the cardinality of the original set $A$.

Accepted Answer

6

Question 20

Given $A = \{1, 2\}$, which of the following are elements of the power set $P(A)$?

Answer

\emptyset

Answer

\{1\}

Answer

\{2\}

Answer

\{1, 2\}

Answer

\{1, 2, 3\}

Fundamentals Of Set Theory And Relations Assessment Practice Tests

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Fundamentals of Set Theory and Relations Assessment

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