Multiple Choice

1. Let $A = \{x \mid x \text{ is a distinct letter in the word 'MATHEMATICS'}\}$. What is the cardinal number of set $A$, denoted as $n(A)$?

   $8$

   $7$

   $11$

   $9$

2. Which of the following sets is an empty set?

   The set of whole numbers less than $1$.

   The set of prime numbers that are even.

   The set of integers $x$ such that $x^2 = 4$.

   The set of months in a year with $32$ days.

3. Consider the set $S = \{x \mid x \in \mathbb{N} \text{ and } 5 < x \leq 10\}$, where $\mathbb{N}$ represents the set of natural numbers. Which of the following is the correct roster form for set $S$?

   $\{6, 7, 8, 9, 10\}$

   $\{5, 6, 7, 8, 9\}$

   $\{6, 7, 8, 9\}$

   $\{5, 6, 7, 8, 9, 10\}$

4. Which of the following correctly describes an infinite set?

   The set of all multiples of $10$.

   The set of all students currently enrolled in schools worldwide.

   The set of all integers between $1$ and $100$.

   The set of all prime numbers less than $1,000,000$.

5. Which of these sets is \emph{NOT} a unit set?

   The set of positive integers $x$ such that $x^2 = 1$.

   $\{\{2\}\}$

   $\emptyset$

   $\{1, 1\}$

6. Let $P = \{y \mid y \text{ is an integer and } y^2 = 25\}$. What are the elements of set $P$?

   $\{-5\}$

   $\{5\}$

   $\{-5, 5\}$

   $\{25\}$

7. Given the set $B = \{a, \{b\}, c, \{d, e\}\}$. Which of the following is an element of set $B$?

   $d$

   $\{b\}$

   $\{a\}$

   $b$

8. Let $Q = \{k \mid k \text{ is a prime number and } 20 < k < 30\}$. What is $n(Q)$?

   $4$

   $1$

   $3$

   $2$

9. Which set-builder notation correctly represents the set $F = \{1, 8, 27, 64, 125\}$?

   $\{n^2 \mid n \in \mathbb{N} \text{ and } 1 \leq n \leq 5\}$

   $\{n^3 \mid n \in \mathbb{Z} \text{ and } 0 \leq n \leq 4\}$

   $\{n^3 \mid n \in \mathbb{N} \text{ and } n < 5\}$

   $\{n^3 \mid n \in \mathbb{N} \text{ and } 1 \leq n \leq 5\}$

10. Which of the following is NOT an empty set?

    The set of natural numbers $x$ such that $x+1 = 0$.

    The set of even prime numbers greater than $2$.

    $\{\emptyset\}$

    The set of all triangles with five sides.

11. Which set represents all real numbers $x$ that are greater than or equal to $-3$ and strictly less than $5$?

    $\{x \mid x \in \mathbb{R}, -3 \leq x < 5\}$

    $\{x \mid x \in \mathbb{R}, -3 < x \leq 5\}$

    $\{x \mid x \in \mathbb{R}, -3 < x < 5\}$

    $\{x \mid x \in \mathbb{R}, -3 \leq x \leq 5\}$

12. Let $C = \{y \mid y \text{ is a multiple of } 4 \text{ and } 15 < y \leq 30\}$. What is $n(C)$?

    $3$

    $2$

    $5$

    $4$

13. Which of the following is an infinite set?

    The set of all points on a given line.

    The set of all letters in the English alphabet.

    The set of all days in the past $1000$ years.

    The set of all positive integers less than $1,000,000$.

14. Given the set $K = \{p, \{q, r\}, s, t\}$. How many elements does set $K$ have?

    $2$

    $3$

    $4$

    $5$

15. Consider the set $H = \{1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots\}$. Which set-builder notation best describes $H$?

    $\{\tfrac{1}{n} \mid n \in \mathbb{Z} \text{ and } n \neq 0\}$

    $\{\tfrac{1}{n} \mid n \in \mathbb{N}\}$

    $\{\tfrac{1}{n} \mid n \in \mathbb{R} \text{ and } n \neq 0\}$

    $\{n \mid n \in \mathbb{N}\}$

16. Let $Z = \{m \mid m \text{ is an integer and } m^2 = 7\}$. What is the cardinal number of set $Z$?

    $1$

    $2$

    Undefined

    $0$

17. Which set is equivalent to the set of all prime numbers between $10$ and $20$?

    $\{11, 13, 17, 19\}$

    $\{11, 13, 15, 17, 19\}$

    $\{11, 13, 17\}$

    $\{13, 17, 19\}$

18. Which of the following sets is a finite set?

    The set of all multiples of $5$ between $1$ and $500$.

    The set of all prime numbers.

    The set of all integers.

    The set of all rational numbers.

19. Let $D = \{z \mid z \in \mathbb{Z}, -5 < z \leq -1\}$. How many elements are in set $D$嶼?

    $5$

    $4$

    $2$

    $3$

20. Consider the set $E = \{x \mid x \in \mathbb{N} \text{ and } x^2 + 4 = 0\}$. What kind of set is $E$?

    A finite set with more than one element

    An infinite set

    A unit set

    An empty set