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Quantitative Aptitude · Sets, Relations and Functions, Limits and Continuity

Sets and Their Representation for CA Foundation

Updated 1 October 2026 · Fact-checked

A set is a well-defined collection of distinct objects. You write it in roster form (list the elements) or set-builder form (state a rule). Most MCQs test element counting, types of sets, and the formulas: a set with n elements has 2ⁿ subsets, 2ⁿ − 1 proper subsets and a power set of 2ⁿ elements.

Understand Sets and Their Representation

A set is a well-defined collection of distinct objects. "Well-defined" means you can say clearly whether any object belongs to it or not. "The set of prime numbers below 10" is well-defined. "The set of tall students" is not, because tall has no fixed meaning.

The objects are called elements or members. We write 3 ∈ A when 3 belongs to A, and 4 ∉ A when it does not. Sets are named with capital letters. Order does not matter and repeats are ignored, so {1, 2, 3} and {3, 1, 2, 2} are the same set.

There are two ways to write a set. In roster (tabular) form you list the elements inside braces, separated by commas: A = {2, 3, 5, 7}. In set-builder form you give the rule: A = {x : x is a prime number, x < 10}. The symbol ":" or "|" reads as "such that".

Types of sets. An empty (null) set has no element, written ∅ or { }. A singleton has exactly one element. A finite set has a countable number of elements; an infinite set does not end. Equal sets have exactly the same elements. Equivalent sets have the same number of elements, not necessarily the same ones. The universal set U contains all elements under discussion in a problem.

Subsets. A is a subset of B (A ⊆ B) if every element of A is in B. If A ⊆ B and A ≠ B, A is a proper subset (A ⊂ B). The empty set is a subset of every set, and every set is a subset of itself. The power set P(A) is the set of all subsets of A. Its elements are sets, so P(A) always includes ∅ and A itself.

Key formulas to remember

Number of subsets
n(P(A)) = 2ⁿ
n is the number of elements in A. This counts the empty set and A itself.
Number of proper subsets
2ⁿ − 1
Excludes only A itself. The empty set is still counted as a proper subset when A is non-empty.
Number of non-empty proper subsets
2ⁿ − 2
Excludes both ∅ and A.
Subsets of fixed size
Number of subsets with r elements = nCr
Useful when a question asks for subsets with exactly 2 or 3 elements.
Equal sets
A = B if and only if A ⊆ B and B ⊆ A
Order and repetition do not matter.
Empty set rule
∅ ⊆ A for every set A
∅ is not the same as {0} or {∅}. {∅} has one element.

How to solve Sets and Their Representation questions

Use this method for any question on representation, types or subsets of sets.

  1. 1Read the set carefully. If it is in set-builder form, convert it to roster form by testing the values the rule allows.
  2. 2Check every condition in the rule, such as natural number, integer, or an inequality. Missing one condition changes the set.
  3. 3Remove repeated elements and count only distinct ones to get n.
  4. 4Identify the type asked: empty, singleton, finite, equal, equivalent, or subset.
  5. 5If the question asks for subsets, apply 2ⁿ, 2ⁿ − 1 or 2ⁿ − 2 according to the exact wording.
  6. 6For a statement-based question, test each statement with a small example set such as {1, 2}.
  7. 7Match your answer with one option and check the wording once more (proper, non-empty, or exactly r elements).

Quickest way: Count first, then plug into 2ⁿ

When to use it: Use this for MCQs on subsets, power sets and proper subsets, and for true/false statements on sets.

  1. Find n quickly. Convert the rule to a list in your head and count distinct elements.
  2. Remember the powers of 2: 2, 4, 8, 16, 32, 64, 128, 256.
  3. Match the wording: all subsets = 2ⁿ, proper = 2ⁿ − 1, non-empty proper = 2ⁿ − 2.
  4. For true/false options, test with {1, 2}. Its subsets are ∅, {1}, {2}, {1, 2}, which exposes most wrong statements.
  5. If options differ by 1 or 2, you are probably being tested on the word proper or non-empty. Re-read before marking.
  6. Skip a question only if the set rule needs long listing; otherwise it is usually a 30-second question.

Common mistakes in Sets and Their Representation

  • Counting repeated elements while finding n.

    Students count the symbols written instead of the distinct elements.

    Fix: Cross out repeats first. {1, 2, 2, 3} has n = 3, so 2ⁿ = 8.

  • Using 2ⁿ when the question asks for proper subsets.

    The formulas look alike and the word proper is missed.

    Fix: Underline the word proper. Proper subsets = 2ⁿ − 1, because A itself is excluded.

  • Saying ∅ is not a subset, or treating ∅ as {0}.

    Students think a subset must contain something.

    Fix: Remember ∅ ⊆ A for every A. {0} has one element, so it is not empty.

  • Confusing ∈ with ⊆.

    Both mean belonging, so the symbols get mixed.

    Fix: Use ∈ between an element and a set, and ⊆ between two sets. 2 ∈ {1, 2} but {2} ⊆ {1, 2}.

  • Treating equivalent sets as equal sets.

    The names sound similar.

    Fix: Equal means identical elements. Equivalent means only the same count. {1, 2} and {a, b} are equivalent but not equal.

  • Ignoring conditions in set-builder form, such as x being a natural number.

    Students rush and include 0 or negative values.

    Fix: Write down the number system first. Natural numbers start from 1 and integers include negatives.

Worked examples

Example 1

If A = {x : x is a natural number, 2x − 3 < 7}, how many proper subsets does A have? (a) 15 (b) 16 (c) 31 (d) 32

Show the solution
  1. Solve the condition: 2x − 3 < 7 gives 2x < 10, so x < 5.
  2. Natural numbers less than 5 are 1, 2, 3, 4. So A = {1, 2, 3, 4} and n = 4.
  3. Proper subsets = 2ⁿ − 1 = 2⁴ − 1 = 16 − 1 = 15.

Answer: (a) 15

Example 2

Which of the following is a singleton set? (a) {x : x is an even prime number} (b) {x : x is a prime number less than 10} (c) {x : x is an integer, x² = 4} (d) {x : x is a natural number, x < 1}

Show the solution
  1. Option (a): the only even prime is 2, so the set is {2}. It has one element.
  2. Option (b): the set is {2, 3, 5, 7}, with four elements.
  3. Option (c): x² = 4 gives x = 2 or −2, so two elements.
  4. Option (d): no natural number is less than 1, so the set is empty.

Answer: (a) {x : x is an even prime number}

Example 3

A set has 7 elements. How many of its subsets are non-empty and proper? (a) 126 (b) 127 (c) 128 (d) 124

Show the solution
  1. Here n = 7, so the total number of subsets is 2⁷ = 128.
  2. Remove the empty set, which is one subset.
  3. Remove the set itself, which is another subset.
  4. Non-empty proper subsets = 128 − 2 = 126.

Answer: (a) 126

Exam tips

  • Read the exact wording: all subsets, proper subsets, and non-empty proper subsets give three different answers, and wrong options are built around them.
  • In set-builder questions, list the elements first. Most mistakes come from guessing the count without listing.
  • Learn the powers of 2 up to 2¹⁰ = 1,024 so you do not waste time on multiplication.
  • For true/false statements, test with {1, 2} or with ∅ rather than arguing in general.
  • With 0.25 negative marking, guess only after you have removed at least one option using a quick check.

Practice questions from Sets, Relations and Functions, Limits and Continuity

Sets and Their Representation: frequently asked questions

What is the difference between roster form and set-builder form?

Roster form lists every element, for example {1, 2, 3}. Set-builder form states a rule, for example {x : x is a natural number, x < 4}. Both describe the same set. Roster form suits small finite sets, and set-builder suits large or infinite ones.

What is the formula for the number of subsets of a set?

A set with n elements has 2ⁿ subsets. This includes the empty set and the set itself. The number of proper subsets is 2ⁿ − 1.

Is the empty set a subset of every set?

Yes. The empty set has no element that could fail to be in another set, so ∅ ⊆ A is true for every set A. It is a proper subset of every non-empty set.

What is a power set?

The power set of A is the set of all subsets of A, written P(A). If A has n elements, P(A) has 2ⁿ elements. For A = {1, 2}, P(A) = {∅, {1}, {2}, {1, 2}}.