Sets and set operations

Modern mathematics begins with a single quiet idea: collect objects together and reason about the collection as a thing in itself. That collection is called a set, and Class 11 — the first major JEE Main topic — is where it formally enters the syllabus. Master sets cleanly and the whole of Relations, Functions and even Probability becomes far easier later.

Definition: A SET is a well-defined collection of distinct objects. The phrase "well-defined" carries weight — given any object, you must be able to say with certainty yes, it belongs or no, it doesn't. "Distinct" means no element is counted twice.

Notation: a set can be described two ways.

• Roster (tabular) form — list the elements inside curly braces, e.g. A = {1, 2, 3}.
• Set-builder form — describe the rule that elements satisfy, e.g. B = {x : x is an even natural number less than 10}, read as "the set of all x such that x is an even natural number less than 10".

The colon : (or vertical bar |) is read as "such that". Order does not matter (so {1, 2, 3} = {3, 1, 2}), and repetition is ignored (so {1, 1, 2} = {1, 2}).

Key types of sets

A handful of named sets recur throughout every chapter. Learn them by their defining property, not just by their name.

• Empty set ∅ (also called null or void set). It has no elements. By convention it is written as ∅ or { }. Crucially, ∅ is a subset of every set — including itself.
• Singleton set. Exactly one element, like {0} or {π}.
• Finite set. A definite, countable number of elements; you can list them all and stop.
• Infinite set. Cannot be listed exhaustively, e.g. the set of natural numbers ℕ.
• Equal sets. Two sets A and B are equal if every element of A is in B and vice versa. Same elements, regardless of order or repetition.
• Equivalent sets. Same number of elements, not necessarily the same elements. {1, 2, 3} and {a, b, c} are equivalent but not equal.
• Subset. A ⊆ B means every element of A is also in B. We say "A is a subset of B".
• Proper subset. A ⊂ B means A ⊆ B but A ≠ B — A is "strictly smaller" than B.
• Universal set U. The fixed master set that contains all the elements under discussion in a given problem. Choosing U is part of setting up the question; different problems choose different universal sets.
• Power set P(A). The set of all subsets of A — including ∅ and A itself.

Two relationships are always true and worth memorising as reflexes: ∅ ⊆ A and A ⊆ A for every set A.

Counting subsets — the 2ⁿ rule

This is the single highest-yield numerical skill in the chapter.

Result: If a set A has n elements, then it has exactly 2ⁿ subsets, so |P(A)| = 2ⁿ.

Why? For each element of A, there are exactly two independent choices when forming a subset: include it, or leave it out. With n elements, that gives 2 × 2 × … × 2 = 2ⁿ possibilities. Every distinct sequence of in/out choices produces a distinct subset; nothing is double-counted; ∅ corresponds to "leave them all out" and A itself corresponds to "include them all".

From this, two derived counts:

• Number of proper subsets = 2ⁿ − 1 (we remove the case where every element is included, which is A itself).
• Number of non-empty subsets = 2ⁿ − 1 (we remove the empty subset).

Notice the trap: both expressions are 2ⁿ − 1, but they correspond to different deletions. If the question asks for proper non-empty subsets, the answer is 2ⁿ − 2.

Worked example — power set count

Question: If A = {a, b, c, d}, find (i) the number of subsets, (ii) the number of proper subsets, (iii) the number of proper non-empty subsets.

Solution:
Step 1: n = 4.
Step 2: Total subsets = 2⁴ = 16.
Step 3: Proper subsets = 2⁴ − 1 = 15 (drop A itself).
Step 4: Proper non-empty subsets = 2⁴ − 2 = 14 (drop A and ∅).
Conclusion: 16, 15 and 14 respectively.

Worked example — set-builder to roster

Question: Write A = {x : x ∈ ℕ, x² ≤ 20} in roster form.

Solution:
Step 1: x must be a natural number with x² ≤ 20.
Step 2: Test x = 1 → 1 ≤ 20 ✓, x = 2 → 4 ✓, x = 3 → 9 ✓, x = 4 → 16 ✓, x = 5 → 25 ✗.
Conclusion: A = {1, 2, 3, 4}.

This conversion skill is tested both ways in JEE Main; expect at least one question that hides a tricky condition (like x ∈ ℤ vs x ∈ ℕ, or strict vs non-strict inequality).

Standard number sets in JEE

The Indian Class 11 NCERT fixes a notation you should treat as universal:

• ℕ — natural numbers {1, 2, 3, …}.
• W — whole numbers {0, 1, 2, 3, …}.
• ℤ — integers, both positive and negative including 0.
• ℚ — rationals, p/q form with q ≠ 0.
• ℝ — real numbers.

The inclusion chain is ℕ ⊂ W ⊂ ℤ ⊂ ℚ ⊂ ℝ, with each step strict.

Why it matters: every later topic — Relations, Functions, Probability, Limits — is phrased in the language of sets. Domain and range are sets. Events in probability are subsets of a sample space. Even definitions in topology and analysis trace back to subset language. Getting the basics into your muscle memory in Class 11 pays interest for years.

Real-world example: every WhatsApp group is a set. Members are elements; "WhatsApp groups inside a college" form a power-set-like collection of subsets of the larger student set. The phrase "you must be in the college to join the cricket group" is the subset relation in everyday language: cricket group ⊆ college.

Common misconception: students often think ∅ is "nothing" and so cannot be a subset of any set. In fact ∅ is a perfectly legal set (the set with no elements), and the statement "every element of ∅ is in A" is vacuously true — there are no elements in ∅ that could possibly violate the condition. Hence ∅ ⊆ A for every set A.

Core operations:

• Union A ∪ B: in A OR B.
• Intersection A ∩ B: in A AND B.
• Difference A − B: in A but NOT B.
• Complement A' = U − A.
• Symmetric difference A Δ B = (A−B) ∪ (B−A) = (A∪B) − (A∩B).

Key identities:

• Commutative, Associative for ∪ and ∩.
• Distributive: A ∩ (B ∪ C) = (A∩B) ∪ (A∩C); A ∪ (B ∩ C) = (A∪B) ∩ (A∪C).
• DE MORGAN'S LAWS (very high-yield): (A ∪ B)' = A' ∩ B'; (A ∩ B)' = A' ∪ B'.
  Memory: 'Break the bar, flip the operation.'
• A − B = A ∩ B'.

Inclusion–Exclusion (counting):
|A ∪ B| = |A| + |B| − |A ∩ B|
|A ∪ B ∪ C| = |A|+|B|+|C| − |A∩B| − |B∩C| − |C∩A| + |A∩B∩C|.

Q: In a class of 100, 60 study Maths, 45 study Physics, 30 study both. How many study (a) at least one, (b) neither, (c) exactly one?

Let |M|=60, |P|=45, |M∩P|=30, |U|=100.

(a) At least one = |M ∪ P| = |M| + |P| − |M∩P| = 60 + 45 − 30 = 75.

(b) Neither = |U| − |M ∪ P| = 100 − 75 = 25.
Using De Morgan: |M' ∩ P'| = |(M ∪ P)'| = 100 − 75 = 25. ✓

(c) Exactly one = |M ∪ P| − |M ∩ P| = 75 − 30 = 45.
Or: (only M) + (only P) = (60−30) + (45−30) = 30 + 15 = 45. ✓

Exam tip: Draw a Venn diagram and fill the INTERSECTION first, then subtract to get 'only' regions. Most JEE set problems are inclusion–exclusion with 2 or 3 sets — memorise the 3-set formula's sign pattern (+ singles, − pairs, + triple).

Cover the answer, recall, then check. 12 must-know cards on sets for JEE Main.

Q1. Number of subsets and proper subsets of a set with n elements?
A1. Subsets = 2ⁿ (this is the power set size). Proper subsets = 2ⁿ − 1. Non-empty subsets = 2ⁿ − 1.

Q2. State the two-set inclusion–exclusion formula.
A2. n(A∪B) = n(A) + n(B) − n(A∩B).

Q3. Three-set inclusion–exclusion for n(A∪B∪C)?
A3. n(A)+n(B)+n(C) − n(A∩B) − n(B∩C) − n(C∩A) + n(A∩B∩C).

Q4. De Morgan's laws for complements?
A4. (A∪B)′ = A′∩B′ and (A∩B)′ = A′∪B′.

Q5. n(A − B) in terms of intersection?
A5. n(A − B) = n(A) − n(A∩B). Also A − B = A ∩ B′.

Q6. Symmetric difference A△B and its count?
A6. A△B = (A−B) ∪ (B−A). n(A△B) = n(A) + n(B) − 2·n(A∩B).

Q7. Number of elements in exactly one of A or B?
A7. n(A) + n(B) − 2·n(A∩B) (same as n(A△B)).

Q8. Number of elements in neither A nor B (universe U)?
A8. n(U) − n(A∪B).

Q9. Distributive laws of ∪ over ∩ (and vice versa)?
A9. A∪(B∩C) = (A∪B)∩(A∪C); A∩(B∪C) = (A∩B)∪(A∩C).

Q10. Cardinality of the Cartesian product A×B?
A10. n(A×B) = n(A)·n(B). Also n(A×B) = n(B×A) but A×B ≠ B×A as sets.

Q11. How many relations exist from A to B?
A11. 2^(n(A)·n(B)) — each ordered pair is either in or out of the relation.

Q12. For exactly two of three sets A, B, C, what is the count?
A12. n(A∩B)+n(B∩C)+n(C∩A) − 3·n(A∩B∩C). (Exactly one = Σn(A) − 2Σn(A∩B) + 3n(A∩B∩C).)

Sets are the language of all higher mathematics, and in JEE Main they surface directly (1 question, ~4 marks, most years) plus indirectly through relations, functions, and probability. The scoring sub-skill is Venn-diagram counting with inclusion–exclusion — nearly every direct question is a survey-type "how many liked exactly one product" problem.

Core operations

Union ∪, intersection ∩, difference A−B = A∩B′, complement A′ = U−A, and symmetric difference A△B. A set with n elements has 2ⁿ subsets (the power set) and 2ⁿ−1 proper subsets.

Key counting formulas

Quantity	Formula
n(A∪B)	n(A)+n(B)−n(A∩B)
n(A∪B∪C)	Σn(A) − Σn(A∩B) + n(A∩B∩C)
n(A−B)	n(A) − n(A∩B)
n(A△B)	n(A)+n(B) − 2n(A∩B)
exactly one of A,B	n(A)+n(B) − 2n(A∩B)
number of subsets	2ⁿ
relations from A to B	2^(n(A)·n(B))

Laws to memorise: De Morgan — (A∪B)′=A′∩B′, (A∩B)′=A′∪B′; Distributive — A∪(B∩C)=(A∪B)∩(A∪C).

Exam Tricks & Tips

• 🎯 For three-set word problems, draw the Venn diagram and fill the centre (A∩B∩C) first, then work outward subtracting overlaps — this avoids double counting.
• 🎯 "At least one" = n(A∪B∪C); "none" = n(U) − n(A∪B∪C). Read the wording carefully — it decides the sign.
• 🎯 Max of n(A∩B) is min(n(A), n(B)); min of n(A∪B) is max(n(A), n(B)). Range questions hinge on these bounds.
• 🎯 Number of subsets containing a fixed element = 2ⁿ⁻¹ (half of all subsets).
• 🎯 A − B, B − A and A∩B are pairwise disjoint and partition A∪B — use them to set up equations quickly.
• ❌ Common mistake: writing n(A∪B∪C) with a minus sign on the triple term. It is +n(A∩B∩C) — the inclusion–exclusion sign alternates.

Expected exam pattern

One direct question, almost always a two- or three-set Venn/survey problem asking for "exactly one", "at least two", or "neither". Occasionally a subset-counting or power-set problem. Low difficulty — treat it as a guaranteed mark.

Quick recap

Memorise 2ⁿ subsets, both inclusion–exclusion formulas, De Morgan's laws, and the "exactly one / exactly two" expressions. Draw a Venn diagram, fill the innermost region first, and mind the alternating signs.