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Sets & Relations — NDA Mathematics

Sets & Relations is among the most reliable scoring chapters in NDA Maths — 75 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.

The handout has every subtopic, worked example, formula and trap with answers shown. The formula sheet is the chapter’s formulas and traps alone, on one page.

All 75 past questions NDA Maths strategy NDA home All formulas on one page

Subtopic notes

Formula & revision sheet

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

Set Fundamentals, Operations and Algebra

Formulas (2)

Union, intersection, complement and difference · Difference and complement identities $A - B = A \cap B' \qquad (A')' = A$
Symmetric difference and set-equality conditions · Symmetric difference $A \triangle B = (A - B) \cup (B - A) = (A \cup B) - (A \cap B)$

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

Formulas (4)

Power set and counting subsets · Counting subsets $|P(A)| = 2^n \qquad \text{proper subsets} = 2^n - 1 \qquad \text{subsets containing a fixed element} = 2^{n-1}$
Inclusion-exclusion for two sets · Inclusion–exclusion (two sets) $|A \cup B| = |A| + |B| - |A \cap B| \qquad |A \cap B| \ge |A| + |B| - |U|$
Inclusion-exclusion for three sets · Inclusion–exclusion (three sets) $|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |B \cap C| - |A \cap C| + |A \cap B \cap C|$
Survey problems — exactly one, exactly two, all three · Survey accounting identities $\text{exactly two} = \textstyle\sum\text{pairwise} - 3\,(\text{all three}) \qquad \text{at least two} = \text{exactly two} + (\text{all three})$

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

Formulas (1)

Cartesian product, domain and range · Cartesian product and relation counts $|A \times B| = |A|\,|B| \qquad \text{number of relations} = 2^{|A||B|} \qquad (A \times B) \cap (B \times A) = (A \cap B) \times (A \cap B)$

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

PYQ weightage by concept

12 concepts · 75 PYQs — where the marks actually sit, so you know what to drill first

Set Fundamentals, Operations and Algebra25 PYQs · 33%

Concept	PYQs	Share
Union, intersection, complement and difference	10	13%
The laws of set algebra	7	9%
Sets — notation, empty set, and equal vs equivalent	4	5%
Symmetric difference and set-equality conditions	4	5%

Counting, Subsets and Inclusion-Exclusion28 PYQs · 37%

Concept	PYQs	Share
Power set and counting subsets	11	15%
Inclusion-exclusion for three sets	6	8%
Survey problems — exactly one, exactly two, all three	6	8%
Inclusion-exclusion for two sets	5	7%

Relations and the Cartesian Product22 PYQs · 29%

Concept	PYQs	Share
Cartesian product, domain and range	9	12%
Testing a relation defined by a rule	7	9%
Reflexive, symmetric, transitive, equivalence	3	4%
Inverse relations and combining relations	3	4%

Test yourself on Sets & Relations

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