Finite, Infinite, Countable and Uncountable Sets MCQs: 12 Solved Questions

Finite versus infinite usually feels easy. The trouble begins when an infinite set is assumed to be uncountable, or when a short real interval is assumed to have fewer elements than the integers or rationals. Those mistakes require three different tests: direct counting for finite rosters, an explicit enumeration for countably infinite sets, and an impossibility proof for uncountable sets.

Choose an option first and write one decisive classification or mapping before reading the explanation. If the notation needs a refresh, begin with Set Theory and Relations Explained for GATE, then place this set within the wider GATE CS Exam Preparation Courses & Test Series route. The term "countable" includes finite and countably infinite sets unless a question separates them explicitly.

Finite, countably infinite and uncountable: one worked classification

A finite set has exactly $n$ elements for some non-negative integer $n$. A countably infinite set can be paired one-to-one with the positive natural numbers. An uncountable set cannot be written as $a_1,a_2,a_3,\ldots$ without missing something. This gives a reusable ladder: count first; if the set never ends, try to construct an explicit sequence; if every proposed sequence can be defeated, prove that no enumeration works. Merely failing to find a list is not enough to prove uncountability. The goal is a witness, not a vague impression of size.

Take $F=\{2,4,6,8\}$. Since $|F|=4$, it is finite and therefore countable under our convention. For $E=\{2,4,6,\ldots\}$, the function $f(n)=2n$ pairs every positive natural number with one member of $E$, so $E$ is infinite but countable. The integers are also listable: $0,1,-1,2,-2,3,-3,\ldots$.

Now suppose the reals in $(0,1)$ were listed as $r_1=0.123\ldots$, $r_2=0.314\ldots$, $r_3=0.271\ldots$, and so on. Build $y$ by choosing its $n$th digit as 1 when the $n$th digit of $r_n$ is not 1, and 2 otherwise. Then $y$ differs from every $r_n$ at digit $n$, so the list misses it. Using only 1 and 2 avoids the recurring-9 representation trap. Thus $(0,1)$ is uncountable.

Cardinality and finite-set MCQs 1-3

Question 1

What is the cardinality of a set?

• A. The maximum element in a set

• B. The minimum element in a set

• C. The number of elements in a set

• D. The sum of all elements in a set

Answer: C. The number of elements in a set. Cardinality measures size, not an arithmetic property of the members. For $T=\{red, blue, green\}$, $|T|=3$, although maximum, minimum and sum are not meaningful here.

Question 2

What is the cardinality of the set {1, 2, 3, 4}?

• A. 3

• B. 4

• C. 5

• D. 6

Answer: B. 4. The roster has four distinct members, so $|\{1,2,3,4\}|=4$. Repeated writing does not add members: $|\{1,1,2,2,3,4\}|=4$ as well.

Question 3

Which of the following sets is finite?

• A. Set of all real numbers

• B. Set of all integers

• C. Set of all natural numbers

• D. {1, 2, 3, 4, 5}

Answer: D. {1, 2, 3, 4, 5}. This roster stops at five distinct elements, so its cardinality is 5. The sets $\mathbb R$, $\mathbb Z$ and $\mathbb N$ are infinite. Being countable does not make $\mathbb Z$ or $\mathbb N$ finite.

Infinite does not mean uncountable: MCQs 4-5

Question 4

If A = {x | x is an even natural number}, is A finite or infinite?

• A. Finite

• B. Infinite

• C. Null

• D. Universal

Answer: B. Infinite. Here $A=\{2,4,6,8,\ldots\}$. Any proposed largest member $2k$ is followed by the larger member $2(k+1)$, so the set is not finite. The map $n\mapsto2n$ enumerates it, making it countably infinite.

Question 5

The set A={a | a=√b and b is a prime number} is,

• A. Countable

• B. Uncountable

• C. Finite

• D. Empty

Answer: A. Countable. Read the set as $A=\{\sqrt p\mid p\text{ is prime}\}$, beginning $\sqrt2,\sqrt3,\sqrt5,\sqrt7,\sqrt{11},\ldots$. Indexing the primes as first, second and so on also indexes these distinct roots. The set is countably infinite, not finite or empty.

Countable candidates and finite constructions: MCQs 6-7

Question 6

Which of the following sets is uncountable?

• A. Set of natural numbers

• B. Set of all integers

• C. Set of all positive rational numbers

• D. None of the above

Answer: D. None of the above. Natural numbers already form a list, while integers can be ordered as $0,1,-1,2,-2,\ldots$. Positive rationals can be traversed by diagonals of constant $p+q$, skipping repetitions such as $2/2=1/1$. All three sets are countable.

Question 7

Which of the following are countable? I. A = {x : x is a point on a line} II. B = {x : x ∈ N and x < 100} III. C = number of permutations of the letters of the largest possible English word

• A. Only III

• B. I, III

• C. II, III

• D. I, II, III

Answer: C. II, III. A line has the cardinality of $\mathbb R$, so I is uncountable. Set II has at most 100 members, depending on whether 0 belongs to $\mathbb N$. Statement III is intended as the set of distinct permutations of a fixed finite word; a word of length $m$ has at most $m!$ such permutations. Under that set-valued reading, II and III are finite and countable.

Real intervals and the countable-versus-uncountable split: MCQs 8-10

Question 8

Which of the following sets is uncountable?

• A. Set of all natural numbers

• B. Set of all integers

• C. Set of all real numbers between 0 and 1

• D. {1, 2, 3, 4, 5}

Answer: C. Set of all real numbers between 0 and 1. The natural numbers and integers are countable, and the roster has five members. The diagonal construction above produces a real missing from every proposed list of $(0,1)$, proving that interval is uncountable.

Question 9 (multiple-select)

Which of the following is/are true:

• A. The set of real number is countable

• B. The set of rational numbers is countable

• C. The set of integers is uncountable

• D. The set of real number between 0 to ¼ is uncountable

Answers: B and D. Reduced rational pairs can be traversed diagonally, so $\mathbb Q$ is countable, while $0,1,-1,2,-2,\ldots$ enumerates $\mathbb Z$. The reals are uncountable. The bijection $f(x)=x/4$ sends $(0,1)$ onto $(0,1/4)$, so even that shorter interval is uncountable.

Question 10

UGC NET 2015, Computer Science, Paper 2 (December)

Which of the following is/are not true ? (a) The set of negative integers is countable. (b) The set of integers that are multiples of 7 is countable. (c) The set of even integers is countable. (d) The set of real numbers between 0 and 1⁄2 is countable.

• A. (a) and (c)

• B. (b) and (d)

• C. (b) only

• D. (d) only

Answer: D. (d) only. Positive $n\mapsto-n$ lists the negative integers. With input from all integers, $n\mapsto7n$ and $n\mapsto2n$ list every negative, zero and positive multiple required in (b) and (c). But $x\mapsto x/2$ is a bijection from $(0,1)$ to $(0,1/2)$, so (d) is the only false statement.

Comparing infinite cardinalities: MCQs 11-12

Question 11 (multiple-select)

S={x | 0 < x < 1, x ∈ R}, where R is the set of real numbers. If |S| = n, Q is the set of rational numbers, N is the set of natural numbers, and C is the set of complex numbers, then which of the following is NOT true?

• A. n = |N|

• B. n = |Q|

• C. n = |R|

• D. n = |C|

Answers: A and B are the statements that are NOT true. Since $S=(0,1)$, $n$ is the continuum cardinality, larger than the countable cardinalities of $\mathbb N$ and $\mathbb Q$. The function $x\mapsto\frac12+\frac{\arctan x}{\pi}$ is a bijection from $\mathbb R$ to $(0,1)$, so $n=|\mathbb R|$. Also, the standard cardinality result $|\mathbb C|=|\mathbb R^2|=|\mathbb R|$ gives $n=|\mathbb C|$.

Question 12

S = {(a,b) | a+b ∈ Q, a,b ∈ R}, where Q is the set of rational numbers and R is the set of real numbers. Which of the following is True about S?

• A. S is countably infinite

• B. S is uncountable

• C. S = R × R (S contains every pair of real numbers)

• D. S is finite

Answer: B. S is uncountable. The injection $t\mapsto(t,-t)$ sends every real $t$ to a distinct pair in $S$, since $t+(-t)=0\in\mathbb Q$. Therefore $S$ contains an uncountable family. Yet $(\sqrt2,0)\notin S$, so $S$ is a proper subset of $\mathbb R\times\mathbb R$, formed by the parallel lines $a+b=q$ for $q\in\mathbb Q$.

Answer map, recurring traps and the next practice route

The compact key is 1-C; 2-B; 3-D; 4-B; 5-A; 6-D; 7-C; 8-C; 9-B,D (multiple-select); 10-D; 11-A,B (multiple-select); 12-B.

Before moving on, reproduce four checks without looking: $|\{1,2,3,4\}|=4$; $0,1,-1,2,-2,\ldots$; the bijection $x\mapsto x/4$ from $(0,1)$ to $(0,1/4)$; and the injection $t\mapsto(t,-t)$ into the set in Question 12.

For broader mixed practice, continue with Set Theory and Relations MCQs: 12 Solved, then use Discrete Mathematics MCQs as the subject practice hub. For a complete Set Theory lesson route, use GATE Guidance by Sanchit Sir. When you want timed mixed practice, move to the GATE Test Series.