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Sets and relations open both the Class 11 book and the JEE Main syllabus, so they are often rushed. That is a mistake: the counting rules return in probability, and the properties of relations return when you study functions. This post works through Venn diagram counting, power sets and Cartesian products, and the four types of relations, with examples you can check by hand.
Where are these topics in the syllabus?
The JEE (Main) 2026 syllabus opens with Unit 1: sets and their representation, union, intersection and complement, power set, "relations, type of relations, equivalence relations" and functions. The CBSE 2026-27 curriculum puts sets, Venn diagrams, Cartesian products and the definition of a relation in Class 11, and "types of relations: reflexive, symmetric, transitive and equivalence relations" in Class 12. It also lists practical problems on union and intersection of two sets among Class 11 topics assessed only formatively. NCERT covers these in Class 11 chapters on sets and relations and functions, and in the Class 12 chapter on relations and functions. Check the documents for your exam year.
How do you count with a Venn diagram?
For two sets, n(A ∪ B) = n(A) + n(B) − n(A ∩ B). The intersection is subtracted because it was counted twice.
Example. In a class of 60, 32 students like maths, 28 like physics and 12 like both. Then 32 + 28 − 12 = 48 like at least one, and 60 − 48 = 12 like neither. Only maths: 32 − 12 = 20. Only physics: 28 − 12 = 16.
For three sets, n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(A ∩ C) + n(A ∩ B ∩ C).
Example. Of 100 students, 45 take physics (P), 40 chemistry (C) and 35 biology (B). 15 take P and C, 10 take C and B, 12 take P and B, and 5 take all three.
Fill the diagram from the centre outwards:
All three: 5.
P and C only: 15 − 5 = 10. C and B only: 10 − 5 = 5. P and B only: 12 − 5 = 7.
Only P: 45 − 10 − 7 − 5 = 23. Only C: 40 − 10 − 5 − 5 = 20. Only B: 35 − 5 − 7 − 5 = 18.
Total in at least one: 5 + 10 + 5 + 7 + 23 + 20 + 18 = 88, so 12 take none.
Check with the formula: 45 + 40 + 35 − 15 − 10 − 12 + 5 = 88.
How many subsets, pairs and relations are there?
A set with n elements has 2ⁿ subsets, so its power set has 2ⁿ elements. For {a, b, c} that is 8, including ∅ and the set itself.
If n(A) = p and n(B) = q, then n(A × B) = pq.
A relation from A to B is any subset of A × B, so there are 2^(pq) relations. With p = 2 and q = 3, A × B has 6 pairs and there are 2⁶ = 64 relations.
How do you test the type of a relation?
For a relation R on a set A:
Reflexive: (a, a) ∈ R for every a in A.
Symmetric: whenever (a, b) ∈ R, also (b, a) ∈ R.
Transitive: whenever (a, b) and (b, c) are in R, also (a, c) ∈ R.
Equivalence: all three.
Example 1. On A = {1, 2, 3}, let R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)}.
Reflexive: all three pairs (a, a) are present.
Symmetric: (1, 2) and (2, 1) are both present.
Transitive: (1, 2) and (2, 1) need (1, 1), which is present; (2, 1) and (1, 2) need (2, 2), also present. No other chains exist.
So R is an equivalence relation, with classes {1, 2} and {3}.
Example 2. On the same set, let S = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)}. S is reflexive. It is not symmetric, because (1, 2) is in S but (2, 1) is not. It is not transitive, because (1, 2) and (2, 3) are in S but (1, 3) is not.
Example 3. On the integers, a R b if a − b is divisible by 3. Then a − a = 0 is divisible by 3 (reflexive); if 3 divides a − b, it divides b − a (symmetric); and (a − b) + (b − c) = a − c (transitive). The equivalence classes are the remainders 0, 1 and 2 on division by 3.
By contrast, "a ≤ b" on the real numbers is reflexive and transitive but not symmetric, since 2 ≤ 3 but 3 is not ≤ 2.
Which mistakes cost the most marks?
Confusing ∅ with {∅}. The first has no elements; the second has one.
Mixing up ∈ and ⊂. 1 ∈ {1, 2}, but {1} ⊂ {1, 2}.
Reading "only A" as "A". "Only A" excludes every overlap.
Assuming symmetric means reflexive. {(1, 2), (2, 1)} on {1, 2} is symmetric but not reflexive.
Missing a chain when testing transitivity. List every pair (a, b), (b, c) systematically.
A practice routine
When you test a relation, say why each property holds or fails. In a classic study, Chi, Bassok, Lewis, Reimann and Glaser (1989) found that students who learned well from worked examples generated many self-explanations relating each step to principles in the text, while weaker students produced few and relied heavily on the examples.
Venn drill (twice a week): three two-set and two three-set problems, always filled from the centre and checked with the formula.
Relation tests (weekly): five relations on {1, 2, 3}; for each property write one sentence of reason or a counterexample.
Link forward: the same counting returns in permutations and combinations, and the same care with definitions in functions, domain and range.
If a definition such as transitivity still feels slippery, the Fermi Concept Library covers every concept in physics, chemistry, biology, mathematics and computing, each with its own lesson, so you can revisit it before the next exercise.
Today, write down three relations on {1, 2, 3}: one that is only reflexive, one that is only symmetric and one that is an equivalence relation.
