Sets Operations

• How to represent sets using roster form, set-builder form, and Venn diagrams — the language of Class 11 Maths Chapter 1.
• The meaning of subset, proper subset, universal set, and power set with cardinality formulas used in JEE Main.
• To perform union (∪), intersection (∩), complement (′), and difference (A − B) on finite sets.
• To apply De Morgan's laws and the inclusion–exclusion principle to solve NCERT and competitive-exam counting problems.

Level 1 — Sets and basic operations

Verbal: A set is a well-defined collection of distinct objects. Two sets are equal if they contain exactly the same elements, regardless of order or repetition in listing.

Symbolic: If A = {1, 2, 3} and B = {2, 3, 4}, then:

• A ∪ B = {1, 2, 3, 4}
• A ∩ B = {2, 3}
• A − B = {1}
• A′ (relative to U) = U − A

Visual:

Cardinality: n(A) counts elements. For finite sets: n(A ∪ B) = n(A) + n(B) − n(A ∩ B).

Level 2 — Power set, De Morgan, and applications

Power set: P(A) is the set of all subsets of A. If n(A) = n, then n(P(A)) = 2ⁿ. Example: A = {a, b} → P(A) = {∅, {a}, {b}, {a, b}} — four subsets.

De Morgan's laws (for universal set U):

• (A ∪ B)′ = A′ ∩ B′
• (A ∩ B)′ = A′ ∪ B′

Disjoint sets: A ∩ B = ∅. Then n(A ∪ B) = n(A) + n(B).

Inclusion–exclusion (three sets): n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(C ∩ A) + n(A ∩ B ∩ C)

JEE tip: When a problem says "at least one" or "neither A nor B", translate to union/complement before counting.

NCERT spotlight — Venn diagrams and practical counting

When NCERT asks how many students study Mathematics or Physics but not both, translate words into set notation before calculating. Only A means A minus (A intersection B). Exactly one subject means (A minus B) union (B minus A), the symmetric difference. For three subjects M, P, C, students in at least two subjects equals n(M intersection P) + n(P intersection C) + n(C intersection M) minus 2 times n(M intersection P intersection C).

Power set in competitive exams: If A has n elements, the number of proper subsets is 2^n minus 1. JEE often combines sets with probability by defining events as subsets of sample space S.

Interval notation on R: The set (-infinity, 2) union (5, infinity) is a subset of real numbers. Union here is set union, not an interval endpoint typo.

In a class of 60 students, 35 study Mathematics, 30 study Physics, and 15 study both. How many study neither?

Step 1 — Let M = set of Maths students, P = set of Physics students.
         n(M) = 35, n(P) = 30, n(M ∩ P) = 15, n(U) = 60.
Step 2 — Students in at least one subject:
         n(M ∪ P) = n(M) + n(P) − n(M ∩ P)
                  = 35 + 30 − 15 = 50.
Step 3 — Neither subject: n(M′ ∩ P′) = n(U) − n(M ∪ P) = 60 − 50 = 10.
Step 4 — Verify: 50 + 10 = 60 ✓

Applications in probability and logic

Sample space S for rolling a die is {1,2,3,4,5,6}. Event A = even numbers = {2,4,6}. Complement A' = {1,3,5}. For two dice, |S| = 36 ordered pairs. Event sum equals 7 is {(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)} — six favourable outcomes. Set notation keeps counting systematic for JEE probability introduction.

Review and practice drill

Review checklist: (1) Translate verbal conditions into union, intersection, complement symbols before arithmetic. (2) For three sets, draw Venn diagram and label regions a through g for disjoint regions method. (3) Verify n(A union B) <= n(A) + n(B) always. (4) Power set cardinality doubles when one element added. Practice: In survey of 100 students, 60 like cricket, 40 like football, 20 like both — find only cricket, only football, neither. Answers: only cricket 40, only football 20, neither 20. These numbers must sum to 100.

• If A = {x : x is an even natural number less than 10}, write A in roster form and find n(P(A)).
• Prove (A ∪ B)′ = A′ ∩ B′ using element method.
• In a survey, 40 like tea, 30 like coffee, 10 like both. How many like at least one?

Open the Practice tab for graded questions on Sets Operations.