Sets

If $A = \{x : x \text{ is a letter in the word 'STATISTICS'}\}$, what is the cardinality of $A$?

Which of the following is a null set?

If $A \subset B$, then $A \cap B$ is equal to:

What is the result of $(A')'$?

Two finite sets $A$ and $B$ have $m$ and $n$ elements respectively. If the total number of subsets of set $A$ is 112 more than the total number of subsets of set $B$, calculate the value of $m + n$.

If $U = \{x : x \in \mathbb{N}, x \leq 10\}$ and $A = \{x : x \text{ is even}\}$, $B = \{x : x \text{ is a multiple of 3}\}$, find $A \cap B$.

Which of the following describes the set $\{x : x \text{ is a real number and } x^2 + 1 = 0\}$?

If $A = \{x : x \in \mathbb{N}, x < 5\}$ and $B = \{x : x \in \mathbb{Z}, x^2 < 5\}$, find $n(A \times B)$.

Let $S = \{1, 2, 3, 4\}$. Let $R = \{(1,2), (2,3)\}$ be a subset of $S \times S$. What is the minimum number of elements that must be added to $R$ to make it a reflexive set on $S$?

If $n(A - B) = 10, n(B - A) = 15,$ and $n(A \cup B) = 35$, find $n(A \cap B)$.

What is the cardinality of the set $A = \{\emptyset, \{\emptyset\}, 1\}$?

If $n(A) = 7$ and $n(B) = 5$, what is the maximum possible value of $n(A \cap B)$?

If $n(A) = 7$ and $n(B) = 5$, what is the maximum possible value of $n(A \cup B)$?

If $A = \{x, y\}$, then the power set $P(A)$ is:

For any three sets $A, B$, and $C$, the set expression $A - (B - C)$ is always equivalent to:

Two finite sets $A$ and $B$ have $n$ and $m$ elements respectively. If the total number of subsets of set $A$ is 48 more than the total number of subsets of set $B$, find the value of $n^2 - m^2$.

If $n(A) = 12, n(B) = 8,$ and $A \cap B = \emptyset$, find $n(A \cup B)$.

Given $n(U) = 700, n(A) = 200, n(B) = 300$ and $n(A \cap B) = 100$, find $n(A' \cap B')$.

Which of the following represents the shaded region of $(A \cup B) - (A \cap B)$ in a Venn Diagram?

If $n(P(A)) = 512$, find $n(A)$.

Identify the roster form of the set $W = \{x : x \in N, x \text{ is a factor of 10}\}$?

Represent the set of natural numbers which are multiples of 10 up to 50 in roster form.

Write the set of digits used in the number 1001 in roster form.

Express the set $V = \{x : x = n^2 - 1, n \in \{1, 2, 3, 4\}\}$ in roster form.

Represent the set $U_1 = \{x : x \text{ is an integer and } x^2 < 10\}$ in roster form.

Express the set $T_1 = \{x : x = \frac{n^2}{n+1}, n \in \{1, 2, 3\}\}$ in roster form.

What is the set-builder form for the set $S_1 = \{x : x \text{ is a month of a year having 30 days}\}$?

Find the roster form of the set $U = \{x : x \in \mathbb{Z}, -2 \le x < 1$ and $x$ is not negative$\}$.

Express $T = \{x : x \in \mathbb{N}, x \text{ is a prime factor of } 105\}$ in roster form.

What is the roster form of $S = \{x : x \in \mathbb{Z}, \frac{6}{x} \in \mathbb{Z}$ and $x > 2\}?$

If $E$ is the set of all even natural numbers, what can we say about the number of elements in $E$?

Categorize the set of all prime numbers between 90 and 100.

A set $D$ contains all the possible rectangular coordinate points $(x, y)$ where both $x$ and $y$ are integers. Is $D$ finite or infinite?

What is the nature of the set of all integers whose absolute value is less than 5?

What is the number of elements in the set $V = \{x \in \mathbb{N} : 3x + 4 = 4\}$?

Determine the nature of the set $A = \{x \in \mathbb{Z} : 2 < x < 3\}$.

How many elements are in the set $T = \{x \in \mathbb{N} : x \text{ is a perfect cube } \le 27\}$?

Let $S = \{x \in \mathbb{Z} : x^2 + y^2 = 1 \text{ for some } y \in \mathbb{Z}\}$. Determine the cardinality of $S$.

If $A = \{1, 2, 3\}$ and $B = \{3, 4, 5\}$, find $n(P(A \cap B))$.

Let $A = \{x \in \mathbb{N} : x \text{ is a multiple of } 6, x < 100\}$ and $B = \{x \in \mathbb{N} : x \text{ is a multiple of } 8, x < 100\}$. Find $n(A \cap B)$.

If $A = \{1, 4, 9, 16\}$ and $B = \{x^2 : x \in \{1, 2, 3, 4\}\}$, what is the relation between $A$ and $B$?

If $A = \{x : x \in \mathbb{N}, x \text{ is a factor of } 4\}$ and $B = \{1, 2, 4\}$, are they equal?

Let $A = \{x : x \text{ is a letter in 'WOLF'}\}$ and $B = \{x : x \text{ is a letter in 'FLOW'}\}$, then:

If $A = \{x : x \in \mathbb{Z}, x^2 = 1\}$ and $B = \{1, -1\}$, identify the relation.

If $A = \{x : x \in \mathbb{Z}, x^2 - 4x + 4 = 0\}$ and $B = \{2\}$, are they equal?

Consider $A = \{x : x \text{ is a letter in 'EAT'}\}$ and $B = \{x : x \text{ is a letter in 'TEA'}\}$. Are they equal?

If $S = \{1, 3, 5, 7\}$ and $T = \{x : x \text{ is an odd natural number } < 8\}$, which is correct?

If $A = \{x \in \mathbb{Z} : x^2 - 4 = 0\}$ and $B = \{x \in \mathbb{Z} : |x| = 2\}$, then which is correct?

Find the set of values of $x$ such that $\{x, x+2\} = \{3, 5\}$.

Let $A = \{x : x = n^2, n \in \mathbb{N}\}$ and $B = \{1, 4, 9, 16, \dots\}$. Are they equal?

If $n(P(A)) = 4$, then $n(A)$ is:

Which of the following is an infinite set that is a subset of the real numbers?

If $A = \{1, 2\}$, then the power set $P(A)$ is:

If $A = \{x : x \text{ is a letter in the word 'MATHEMATICS'}\}$, how many subsets of $A$ are there? (Hint: Find $n(A)$ first)

If $A$ is the set of all divisors of 12 and $B$ is the set of all divisors of 6, which relationship holds?

What is the number of subsets of the set containing the letters of the word 'BEE'?

If $X = \{n^2 + 1 : n \in \{0, 1, 2\}\}$ and $Y = \{1, 2, 5, 10\}$, which is correct?

Let $A = \{1, 2, 3, 4, 5, 6\}$. How many subsets of $A$ contain at least two elements such that the product of all elements in the subset is a multiple of 3?

If $A$ and $B$ are two sets such that $n(A) = 4$, $n(B) = 3$, and $A \cap B = \emptyset$, how many subsets of $A \cup B$ contain at least one element from $A$ and at least one element from $B$?

How many subsets of the set $\{x, y, z\}$ contain the element $x$?

What is the cardinality of the set $Q = \{x : x \in \mathbb{R}, x^2 + 1 = 0\}$?

If $n(A) = m$, find the cardinality of $P(A) \times A$.

Find the cardinality of the set $O = \{x : x \in \mathbb{Z}, |x| \le 2\}$.

If $A = \{1, 2, 3\}$ and $B = \{3, 4, 5\}$, what is $n(A \Delta B)$, where $\Delta$ is the symmetric difference?

Let $A = \{x : x \text{ is a two-digit prime number whose digits are the same}\}$. Find $n(A)$.

If $n(A) = 4$, $n(B) = 5$, and $n(C) = 6$, and they are all disjoint, what is $n(P(A \cup B \cup C))$?

If $A = \{1, \{2, 3\}, 4, \{5, 6, 7\}\}$, find $n(A)$.

Let $A = \{ x : x \text{ is a letter in the word 'MATHEMATICS'} \}$. Find the number of proper subsets of $A$.

If the number of subsets of a set $A$ is $S$, and the number of subsets of its power set $P(A)$ is $2^{16}$, find $n(A)$.

If $n(A \cup B) = 100, n(A \cap B) = 40$ and $n(A) = 70$, find $n(B \setminus A)$.

How many elements are in the power set of $A = \{1, 2, \{3, 4\}\}$?

If $P(A) = P(B)$, then which of the following must be true?

If $n(A) = 4$, then the number of subsets of $A$ having at least one element is:

If $A$ is a set such that $n(P(A)) = 2$, then $A$ is a:

Let $A = \{1, 2, 3, 4, 5\}$. How many elements in $P(A)$ contain the element $1$ but not the elements $2$ and $3$?

Given $n(A) = 4$, what is the ratio of the number of subsets to the number of proper subsets?

If $A = \{1, 2\}$, which of the following is the set $P(P(\emptyset)) \cap P(A)$?

How many elements are in $P(P(\{1, 2, 3\}))$ that contain the set $\{1, 2\}$ as an element?

If $n(A) = 3$, how many elements of $P(A)$ are NOT proper subsets of $A$?

If $S$ is a set with $n$ elements, what is the number of pairs $(A, B)$ such that $A \in P(S)$, $B \in P(S)$, and $A \subseteq B$?

Which of the following always represents the complement of set $A$ relative to $U$?

If $A = \{1, 3, 5\}$ and $U = \{1, 2, 3, 4, 5\}$, what is $U - A'$?

If $A$ is a subset of $U$, then $A - U$ is:

If $U = \{x : x \in \mathbb{N}, x \le 10\}$ and $A = \{x : x > 5\}$, what is $A'$?

If $U = \{x : x \in N, x \le 50\}$ and $A = \{x : x \text{ is a perfect square}\}$, how many elements are in $A'$?

If $A \subset U$, what is $A - U$?

If $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$, $A = \{1, 2, 3, 4\}$, and $B = \{2, 4, 6, 8\}$, what is $(A' \cap B)'$?

Let $U = \mathbb{R}$. If $A = [0, 5]$ and $B = [2, 7]$, find the length of the interval representing $(A \cap B)$.

If $U = \{x \in \mathbb{Z} : x^2 < 10\}$, find $n(P(U))$.

Let $U = \{x \in \mathbb{N} : x \leq 15\}$. Let $A$ be the set of prime numbers and $B$ be the set of even numbers. Find $n(A \cup B)'$.

In a survey of 40 people, 25 like Coffee and 20 like Tea. What is the minimum number of people who must like both, assuming everyone likes at least one?

In a Venn diagram, the complement of the universal set ($U'$) is represented by:

If $A \subset B$, then $n(A \cap B)$ is equal to:

If $n(U) = 60$ and $n(A \cup B)' = 10$, then $n(A \cup B)$ is:

In a survey of 100 students regarding their preference for three soft drinks $X, Y$, and $Z$, it was found that 10 students like none of the three drinks and 10 students like all three. It is also known that 20 students like both $X$ and $Y$, 20 students like both $Y$ and $Z$, and 20 students like both $X$ and $Z$. If 20 students like only drink $X$ and the total number of students who like drink $Y$ is equal to the total number of students who like drink $Z$, how many students like only drink $Y$?

In a sports club with 100 members, 60 play tennis, 40 play golf and 20 play both. How many members play tennis but not golf?

In a Venn diagram of three sets $A, B, C$, the region represented by $(A \cap B) - C$ is the part of the diagram that is:

If $A = \{1, 2, 3, 4, 5\}$ and $B = \{1, 3, 5, 7, 9\}$, how many elements are in $(A \cap B) \times (A \cup B)$?

In a group of 52 people, 16 drink tea but not coffee and 33 drink tea. How many drink coffee but not tea?

The number of non-empty subsets of a set containing 10 elements is: