Set Theory
~11 min read
- What: Sets and operations (union, intersection, complement, difference), cardinality, the inclusion-exclusion formula for 2 and 3 sets, Venn diagrams, and standard CDS word problems on "students who study X but not Y" type.
- Why it matters: CDS papers from 2000 to 2023 average 2–3 questions per sitting — small chapter, formulaic answers.
- Key fact: Inclusion-exclusion for two sets: \(n(A \cup B) = n(A) + n(B) - n(A \cap B)\). For three: \(n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(A \cap C) + n(A \cap B \cap C)\). Memorise both.
Set Theory is one of the most "formulaic" chapters in CDS arithmetic. Two identities — the inclusion-exclusion principle for 2 and 3 sets — solve 80% of the questions. The remaining 20% are definitional (subsets, power sets, complements). Quick to learn, quick to score.
This page is built from CDS Previous Year Questions across 2000–2023. Pair with Number System (set notation for number systems) and Linear Equations (set-based word problems often reduce to systems).
What This Topic Covers
CDS scope: (1) definitions — set, element, subset, proper subset, power set, universal set, empty set; (2) operations — union, intersection, complement, difference, symmetric difference; (3) properties — commutative, associative, distributive, De Morgan's laws; (4) cardinality and the inclusion-exclusion principle; (5) Venn diagrams; and (6) word problems — typically two or three overlapping groups, find the size of one region.
Why This Topic Matters
- 2–3 CDS questions per paper, all using two identities.
- Word problems map directly onto inclusion-exclusion.
- De Morgan's laws appear in statement-verification questions.
Exam Pattern & Weightage
| Year / Paper | No. | Subtopics Tested |
|---|---|---|
| 2007-II | 2 | Subsets, power set |
| 2008-I/II | 2 | Union, intersection, complement |
| 2010-II | 2 | Inclusion-exclusion 2 sets |
| 2011-I/II | 3 | Word problems, complement |
| 2012-II | 2 | Inclusion-exclusion 3 sets |
| 2013-I/II | 3 | Word problems, De Morgan |
| 2014-I/II | 2 | Power set, subset count |
| 2016-I/II | 3 | Inclusion-exclusion, complement |
| 2017-I/II | 3 | Word problems, Venn |
| 2018-I/II | 2 | De Morgan, complement |
| 2019-II | 2 | Word problem 3 groups |
| 2020-I/II | 2 | Mixed |
| 2021-I/II | 2 | Inclusion-exclusion |
| 2023-I | 2 | Mixed |
A set with \(n\) elements has \(2^n\) subsets (including \(\emptyset\) and the set itself) and \(2^n - 1\) proper subsets. CDS asks this almost every year — the answer is always a power of 2 minus possibly 1.
Core Concepts
Basic Definitions
Set: a well-defined collection of distinct objects. Element: a member of a set. Subset: \(A \subseteq B\) iff every element of \(A\) is in \(B\). Proper subset: \(A \subsetneq B\) means \(A \subseteq B\) and \(A \neq B\). Power set: \(P(A)\) is the set of all subsets of \(A\). Universal set: \(U\), the set containing all elements under discussion.
Set Operations
Intersection: \(A \cap B = \{x : x \in A \text{ and } x \in B\}\)
Complement: \(A' = \{x \in U : x \notin A\}\)
Difference: \(A - B = \{x : x \in A,\, x \notin B\}\)
Symmetric difference: \(A \triangle B = (A - B) \cup (B - A)\)
De Morgan's Laws
Inclusion-Exclusion Principle
"Number of students who study only \(A\) (not \(B\))" is \(n(A) - n(A \cap B)\), not \(n(A)\). CDS plants this distinction in nearly every word problem.
Standard Set Identities
Commutative: \(A \cup B = B \cup A\), \(A \cap B = B \cap A\). Associative: \((A \cup B) \cup C = A \cup (B \cup C)\). Distributive: \(A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\).
Worked Examples
Example 1 — Subset Count (2014-I)
Q: How many subsets does the set \(\{1, 2, 3, 4, 5\}\) have?
- \(|A| = 5\). \(|P(A)| = 2^5 = 32\) subsets (including \(\emptyset\) and \(A\) itself).
Example 2 — Inclusion-Exclusion 2 Sets (2010-II)
Q: In a class, 30 students play cricket, 20 play football, and 10 play both. How many play at least one?
- \(n(C \cup F) = n(C) + n(F) - n(C \cap F) = 30 + 20 - 10 = 40\).
Example 3 — Only \(A\), Not \(B\) (2013-I)
Q: In a survey, 40 like tea, 30 like coffee, 15 like both. How many like only tea?
- Only tea = \(n(T) - n(T \cap C) = 40 - 15 = 25\).
Example 4 — Three Sets (2012-II)
Q: In a class, 50 students study English, 40 study Hindi, 30 study Sanskrit. 20 study E and H, 15 study H and S, 10 study E and S, 5 study all three. How many study at least one language?
- Apply 3-set inclusion-exclusion: \(50 + 40 + 30 - 20 - 15 - 10 + 5 = 80\).
Example 5 — Complement (2008-II)
Q: If \(U = \{1, 2, \ldots, 10\}\) and \(A = \{2, 4, 6, 8\}\), find \(A'\).
- \(A' = U - A = \{1, 3, 5, 7, 9, 10\}\).
Example 6 — De Morgan (2018-I)
Q: Prove \((A \cup B)' = A' \cap B'\) for \(A = \{1, 2, 3\}\), \(B = \{2, 3, 4\}\), \(U = \{1, 2, 3, 4, 5\}\).
- \(A \cup B = \{1, 2, 3, 4\}\); \((A \cup B)' = \{5\}\).
- \(A' = \{4, 5\}\); \(B' = \{1, 5\}\); \(A' \cap B' = \{5\}\).
- Both equal \(\{5\}\). ✓
Example 7 — Word Problem Three Groups (2019-II)
Q: Of 200 students, 110 study Math, 90 Physics, 80 Chemistry. 50 study Math and Physics, 40 Physics and Chem, 30 Math and Chem, 20 all three. How many study none?
- Apply inclusion-exclusion: \(n(\text{at least one}) = 110 + 90 + 80 - 50 - 40 - 30 + 20 = 180\).
- None = total − at-least-one = \(200 - 180 = 20\).
How CDS Tests This Topic
Five archetypes: (1) count subsets of a given set, (2) two-set inclusion-exclusion word problem, (3) three-set inclusion-exclusion word problem, (4) De Morgan verification or application, (5) complement / difference computation. Each takes 30–60 seconds.
Exam Shortcuts (Pro-Tips)
Shortcut 1 — Subset Counts
\(n\) elements → \(2^n\) total subsets; \(2^n - 1\) non-empty; \(2^n - 1\) proper (depending on definition); \(\binom{n}{k}\) subsets of size \(k\).
Shortcut 2 — Venn Diagram Setup
For 3 overlapping sets, draw three circles. Fill in the central "all three" region first, then "two but not third", then "one only". Sum check.
Shortcut 3 — Total − None Trick
"Students who study none of X, Y, Z" = (Total) − \(n(X \cup Y \cup Z)\). Always compute the union by inclusion-exclusion, subtract from total.
Shortcut 4 — De Morgan Reflex
\((A \cup B)' = A' \cap B'\) and vice versa. Useful when a statement involves a complement of union or intersection.
Shortcut 5 — Symmetric Difference
\(n(A \triangle B) = n(A) + n(B) - 2 n(A \cap B)\). This is \(n(A \cup B) - n(A \cap B)\). Appears in less common CDS questions.
Common Question Patterns
Pattern 1 — Subset Counting
"How many subsets / proper subsets / subsets of size \(k\) does this set have?" Apply \(2^n\), \(2^n - 1\), or \(\binom{n}{k}\).
Pattern 2 — Two-Set Inclusion-Exclusion
"Students who like at least one of X or Y" given counts and overlap. Apply the 2-set formula.
Pattern 3 — Three-Set Inclusion-Exclusion
Three groups, all pairwise overlaps and triple overlap given. Apply the 3-set formula.
Pattern 4 — Only \(A\), Only \(B\), Only \(C\)
"Only \(A\)" = \(n(A) - n(A \cap B) - n(A \cap C) + n(A \cap B \cap C)\). Useful in 3-set problems asking the exact-one or exact-two count.
Pattern 5 — Complement and Difference
"Students who don't like X" = Total − \(n(X)\). "Students who like X but not Y" = \(n(X) - n(X \cap Y)\).
Preparation Strategy
Week 1. Memorise the inclusion-exclusion formulas for 2 and 3 sets. Drill 15 word problems mapping student-survey data onto the formulas. Practice "only A", "only A and B", "all three" extractions.
Week 2. Subset counts, De Morgan's laws, complement and difference operations. Practice Venn-diagram drawing as a visual aid.
Mock testing. Use CDS mock tests. The chapter is fast — slip-ups come from misreading the question (e.g. "exactly one" vs "at least one"). Drill careful reading.
Drill Set Theory Word Problems
CDS mocks with 2-set and 3-set inclusion-exclusion problems. Five archetypes — two key formulas — reflex.
Start Free Mock TestFrequently Asked Questions
How many subsets does a set with \(n\) elements have?
\(2^n\), including the empty set and the set itself. A 5-element set has \(2^5 = 32\) subsets. The number of proper subsets is \(2^n - 1\) (excluding the set itself).
What is the inclusion-exclusion formula for two sets?
\(n(A \cup B) = n(A) + n(B) - n(A \cap B)\). Add the individual counts and subtract the overlap to avoid double-counting.
What is the inclusion-exclusion formula for three sets?
\(n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(A \cap C) + n(A \cap B \cap C)\). Add individuals, subtract pairwise overlaps, add back triple overlap.
What are De Morgan's laws?
\((A \cup B)' = A' \cap B'\) and \((A \cap B)' = A' \cup B'\). Complement of a union is the intersection of complements; complement of an intersection is the union of complements.
How do I count "only \(A\)" or "exactly one of three"?
"Only \(A\)" = \(n(A) - n(A \cap B) - n(A \cap C) + n(A \cap B \cap C)\). "Exactly one" sums "only \(A\)" + "only \(B\)" + "only \(C\)". Draw a Venn diagram to keep regions clear.
How do I find "students who study none"?
Total − \(n(\text{at least one}) = \text{Total} - n(A \cup B \cup C)\). Compute the union via inclusion-exclusion, subtract from total.
Which CDS Maths topics connect to Set Theory?
Number System — sets of \(\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}\). Linear Equations — set word problems often reduce to systems. Probability (in some exams) uses set notation directly.