NDA Maths · Teaching notes

Sets & Relations — NDA Mathematics

Sets & Relations is among the most reliable scoring chapters in NDA Maths — 69 PYQs across 2017–2026, only ~13% HARD, and built on a small number of repeatable techniques. The chapter teaches in three movements, ordered so each builds on the last: (1) Set fundamentals and algebra — what a set is, the operations (union, intersection, complement, difference, symmetric difference), and the laws (distributive, De Morgan, absorption) that drive the bank's signature 'which identity is NOT correct' questions; (2) Counting and inclusion-exclusion — power sets and subset counting, the two- and three-set inclusion-exclusion formulas, and the Venn 'survey' word problems (exactly one / exactly two / at least two / all three) that are the chapter's highest-yield HARD genre; (3) Relations — the Cartesian product, the reflexive / symmetric / transitive / equivalence properties, and the dominant skill of testing those properties on a relation defined by a rule (factor the equation, then check). 12 concepts, every PYQ tagged. The bank rewards method over memory here — learn the handful of techniques and the marks follow.

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Subtopic notes

PYQ weightage by concept

12 concepts · 69 PYQs — where the marks actually sit, so you know what to drill first

Set Fundamentals, Operations and Algebra23 PYQs · 33%

Concept	PYQs	Share
Union, intersection, complement and difference	9	13%
The laws of set algebra	7	10%
Symmetric difference and set-equality conditions	4	6%
Sets — notation, empty set, and equal vs equivalent	3	4%

Counting, Subsets and Inclusion-Exclusion27 PYQs · 39%

Concept	PYQs	Share
Power set and counting subsets	11	16%
Inclusion-exclusion for three sets	6	9%
Survey problems — exactly one, exactly two, all three	6	9%
Inclusion-exclusion for two sets	4	6%

Relations and the Cartesian Product19 PYQs · 28%

Concept	PYQs	Share
Cartesian product, domain and range	7	10%
Testing a relation defined by a rule	6	9%
Reflexive, symmetric, transitive, equivalence	3	4%
Inverse relations and combining relations	3	4%

Formula & revision sheet

0 formulas · 2 reference tables · 13 gotchas across all subtopics — the exam-eve cheat-sheet

Set Fundamentals, Operations and Algebra

Reference tables (1)

The laws of set algebra4 rows

Law	Statement	The impostor to watch for
Distributive	$A\cup(B\cap C)=(A\cup B)\cap(A\cup C)$	Swapping $\cup$ / $\cap$ on one side breaks it
De Morgan	$(A\cup B)'=A'\cap B'$	$(A\cup B)' = A'\cup B'$ is WRONG — the operation flips In predicate form: $x\notin(A\cup B)\Rightarrow x\notin A$ AND $x\notin B$ (not OR).
Absorption	$A\cup(A\cap B)=A$	$A\cup(A\cap B)=A\cup B$ is WRONG — it collapses to just $A$
Subset test 'for all B'	$(A\cap B)\subseteq(C\cap B)\ \forall B \Rightarrow A\subseteq C$	Test such claims by choosing $B=\emptyset$ or $B=E$

Distractors are genuine laws with one operation flipped. Verify a suspect identity on a tiny example or a Venn diagram.

Watch out for (5)

Equal vs equivalent
→ Sets — notation, empty set, and equal vs equivalent
$A - B$ is not symmetric
→ Union, intersection, complement and difference
The three disjoint pieces rebuild the union
→ Union, intersection, complement and difference
The wrong option is a real law with a flipped operation
→ The laws of set algebra
You cannot cancel sets like numbers
→ Symmetric difference and set-equality conditions

Counting, Subsets and Inclusion-Exclusion

Watch out for (4)

Count the elements before raising 2 to a power
→ Power set and counting subsets
Overlap of multiples uses LCM, not product
→ Inclusion-exclusion for two sets
Mind the alternating signs
→ Inclusion-exclusion for three sets
'Exactly two' is not the sum of pairwise intersections
→ Survey problems — exactly one, exactly two, all three

Relations and the Cartesian Product

Reference tables (1)

Reflexive, symmetric, transitive, equivalence4 rows

Property	Test	Fails when
Reflexive	Is $(a,a)\in R$ for every a?	One element lacks its self-loop (e.g. $(4,4)\notin R$ )
Symmetric	Does $(a,b)\in R$ force $(b,a)\in R$ ?	Some arrow has no reverse
Transitive	Do $(a,b),(b,c)$ force $(a,c)$ ?	A 2-step path with no direct shortcut
Equivalence	All three hold	Any one of R / S / T fails A strict inequality $<$ is transitive ONLY — not reflexive, not symmetric.

Test each property separately; one counterexample kills it.

Watch out for (4)

Range can be smaller than the codomain
→ Cartesian product, domain and range
Reflexive means EVERY element, not just some
→ Reflexive, symmetric, transitive, equivalence
Factor before you test
→ Testing a relation defined by a rule
Every function is a relation, not the reverse
→ Inverse relations and combining relations