Quiz: Set Theory Fundamentals — 19 questions

1. Which description best defines a set in mathematics?

A. A statement that compares two mathematical quantities

B. A collection of clearly distinguished objects treated as a whole

C. A rule that assigns one output to each possible input

D. A sequence of numbers arranged according to a fixed calculation

Answer: (B)

Explanation

A set is formed from clearly distinguished objects considered together as one collection. A sequence, function, or comparison statement has a different mathematical purpose.

2. If $a \in A$ is true, what does it tell you about $a$?

A. The object a contains every element of set A

B. The object a is larger than every element of set A

C. The object a belongs to set A

D. The object a is separate from set A

Answer: (C)

Explanation

The symbol $\in$ states that an object belongs to a set. The notation $a \notin A$ would instead indicate that a does not belong to A.

3. Which representation gives a set by stating the condition its elements satisfy?

A. Sequential form

B. Descriptive form

C. Numerical form

D. Roster form

Answer: (B)

Explanation

Descriptive form specifies a set through a condition shared by its elements. Roster form differs because it lists the elements themselves.

4. Why are the sets $\{1,2,3\}$ and $\{3,1,2\}$ equal?

A. They contain the same number of written symbols

B. They were produced using the same description

C. They contain their elements in the same order

D. They contain exactly the same elements

Answer: (D)

Explanation

Set equality depends on having exactly the same elements, not on their order. The displayed order changes, but the membership of the two sets does not.

5. What distinguishes a finite set from an infinite set?

A. An infinite set must contain more than one kind of object

B. A finite set has a listing that eventually stops

C. An infinite set contains elements written in alphabetical order

D. A finite set contains elements of just one type

Answer: (B)

Explanation

A finite set can be listed with a final element, so its listing eventually stops. An infinite set has a listing that continues without reaching a final element.

6. Which classification correctly describes the natural divisors and natural multiples of 6?

A. $T_6=\{1,2,3,6\}$ is infinite, while $V_6=\{6,12,18,\dots\}$ is finite

B. Both $T_6$ and $V_6$ are infinite because both involve natural numbers

C. $T_6=\{1,2,3,6\}$ is finite, while $V_6=\{6,12,18,\dots\}$ is infinite

D. Both $T_6$ and $V_6$ are finite because each begins with 6

Answer: (C)

Explanation

The divisors of 6 form a completed list, whereas the multiples of 6 continue indefinitely. The fact that both sets involve natural numbers does not give them the same size classification.

7. What does $A\subset B$ assert about two sets?

A. The two sets contain exactly the same elements

B. Every element of B also belongs to A

C. Every element of A also belongs to B

D. At least one element of A lies outside B

Answer: (C)

Explanation

The notation $A\subset B$ means that A is contained in B, so each element of A belongs to B. Having an element of A outside B would instead show that A is not a subset of B.

8. Which statement about the empty set is correct?

A. It contains elements that cannot be listed individually

B. It contains one unspecified element and belongs to no set

C. It contains no elements and is a subset of every set

D. It contains every possible element of the universal set

Answer: (C)

Explanation

The empty set has no elements, and the condition for being a subset is therefore satisfied for every set. A set with an unspecified or unlisted element is not necessarily empty.

9. Which elements belong to the intersection $A\cap B$?

A. Elements in the universal set that are outside A

B. Elements that belong to A or B or both

C. Elements that belong to A but not B

D. Elements that belong to both A and B

Answer: (D)

Explanation

The intersection collects elements shared by both sets. The union is broader because it includes elements belonging to either set, including those found in just one.

10. If $A=\{1,2,3\}$ and $B=\{3,4,5\}$, what is $A\cup B$?

A. $\{1,2\}$

B. $\{4,5\}$

C. $\{3\}$

D. $\{1,2,3,4,5\}$

Answer: (D)

Explanation

The union contains every distinct element from either set, so the shared element is listed once along with the other elements. The set $\{3\}$ represents the intersection instead.

11. If $A=\{2,4,6,8\}$ and $B=\{4,8,10\}$, what is $A\setminus B$?

A. $\{2,6\}$

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

C. $\{10\}$

D. $\{4,8\}$

Answer: (A)

Explanation

The difference keeps elements of A that are absent from B, leaving 2 and 6. The set $\{4,8\}$ contains the elements common to both sets and therefore describes their intersection.

12. Let the universal set be $G=\{1,2,3,4,5,6\}$ and $A=\{2,4,6\}$. What is the complement of A in G?

A. $\{4,6\}$

B. $\{1,3,5\}$

C. $\{2,4,6\}$

D. $\{1,2,3,4,5,6\}$

Answer: (B)

Explanation

The complement contains elements of the universal set that do not belong to A, namely 1, 3, and 5. The set $\{2,4,6\}$ is A itself, not its complement.

13. Which expression correctly describes the result of taking the difference of a set with itself?

A. $A\setminus A=A\cap A$

B. $A\setminus A=G$

C. $A\setminus A=A$

D. $A\setminus A=\varnothing$

Answer: (D)

Explanation

Removing every element of A from A leaves no elements, so the result is the empty set. Self-intersection and self-union instead reproduce A, which makes those identities different.

14. Which identity illustrates absorption in set algebra?

A. $A\cap(A\cup B)=B$

B. $A\cap(A\cup B)=A$

C. $A\cup(A\setminus B)=B$

D. $A\setminus(A\cup B)=A$

Answer: (B)

Explanation

The set A already contains every element that can remain in the intersection, so combining it with A union B reduces back to A. The expression involving a union with a difference also relates to absorption, but its result is not B.

15. If $|A|=12$, $|B|=9$, and $|A\cap B|=4$, what is $|A\cup B|$?

A. 17

B. 21

C. 4

D. 8

Answer: (A)

Explanation

The union count is found by adding the two set sizes and subtracting the overlap: $|A\cup B|=12+9-4=17$. Adding without subtracting would count the shared elements twice.

16. If $|A|=15$ and $|A\cap B|=6$, what is $|A\setminus B|$?

A. 6

B. 9

C. 21

D. 15

Answer: (B)

Explanation

The difference removes the six elements shared with B from A, giving $|A\setminus B|=15-6=9$. Adding the overlap would describe a different counting operation rather than removal.

17. A survey counts people who satisfy at least one of two conditions represented by sets A and B. Which procedure avoids double-counting people who satisfy both conditions?

A. Subtract $|B|$ from $|A|$, then add $|A\cap B|$

B. Add $|A|$ and $|B|$, then add $|A\cap B|$

C. Count $|A\cap B|$ and treat it as the total

D. Add $|A|$ and $|B|$, then subtract $|A\cap B|$

Answer: (D)

Explanation

Satisfying at least one condition corresponds to the union, whose count is $|A\cup B|=|A|+|B|-|A\cap B|$. The overlap must be subtracted because those people were included in both individual counts.

18. How is a set in descriptive form converted into roster form?

A. List every object that satisfies the stated condition

B. Arrange the objects according to their numerical size

C. Replace each object with a symbol for its condition

D. State the common property shared by the listed objects

Answer: (A)

Explanation

Roster form is obtained by identifying all objects that satisfy the condition and listing those objects. The common-property description belongs to descriptive form, so it does not perform the requested conversion.

19. Which descriptive form represents the set $\{1,3,7,21\}$?

A. The set of natural multiples of 21

B. The set of natural square numbers

C. The set of odd natural numbers below 22

D. The set of natural divisors of 21

Answer: (D)

Explanation

Each element in the set divides 21, so the set is the collection of natural divisors of 21. Natural multiples of 21 would include numbers such as 21 and 42 rather than the listed divisor pattern.