Chapter 7 of 19

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Andhra Pradesh (BIEAP)Class 10 Mathematics★ AP Board (BSEAP) SSC Mathematics · ~4 marks · Board: AP Board

Sets

SET notation, types of sets, subsets, Venn diagrams, and set operations — union, intersection, difference. AP Board Class 10 Mathematics.

⏱ 8 min readEasy difficulty✓ Free · NCERT-aligned

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Study Guide

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Key points, glossary and the questions most likely to show up.

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By the end of this chapter you'll be able to…

1Use correct set notation: element (∈), empty set (∅), subset (⊆), proper subset (⊂), cardinality n(A)
2Identify types of sets: empty, finite, infinite, equal, equivalent
3Perform set operations: union (∪), intersection (∩), difference (−), complement (A')
4Apply the formula n(A∪B) = n(A) + n(B) − n(A∩B) for word problems
5Draw and interpret Venn diagrams for two and three sets

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Why this chapter matters

Sets is a concise chapter that introduces the language and notation of modern mathematics. Venn diagrams and the cardinal number formula n(A∪B) = n(A)+n(B)−n(A∩B) are the most commonly tested concepts in AP SSC. This formula is used to solve word problems about surveys and classifications. Sets notation (∪, ∩, ∈, ⊆) is foundational for Class 11 mathematics. This is a quick-win chapter for AP SSC students.

Before you start — revise these

A 5-minute refresher here will save you 30 minutes of confusion below.

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Sets

Basic Definitions

Set: Well-defined collection of objects. Element: Member of a set. a ∈ A.

Types of Sets

Type	Definition
Empty (Null)	No elements. ∅ or {}
Finite	Countable number of elements
Infinite	Unlimited elements
Equal	Exactly the SAME elements

Subsets: A ⊆ B if every element of A is in B. Proper subset: A ⊂ B (A ⊆ B and A ≠ B).

Set Operations

Union A ∪ B: Elements in A OR B (or both)
Intersection A ∩ B: Elements in BOTH A and B
Difference A − B: Elements in A but NOT in B
Complement A′: Elements NOT in A

Cardinal Number: n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

Venn Diagrams — Draw overlapping circles to visualise set relationships.

Detailed Set Operations

Union of Sets (A ∪ B)

The set of ALL elements that belong to A OR B (or both).

Example 1: If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, then A ∪ B = {1, 2, 3, 4, 5, 6} ✓ 'List each element only ONCE, even if it appears in both sets.'

Intersection of Sets (A ∩ B)

The set of elements common to BOTH A and B.

Example 2: Using the same A and B: A ∩ B = {3, 4} ✓

Difference of Sets (A − B)

The set of elements in A but NOT in B.

Example 3: A − B = {1, 2} (elements in A that are NOT in B) B − A = {5, 6} (elements in B that are NOT in A)

IMPORTANT: A − B ≠ B − A (set difference is NOT commutative)

Complement of a Set (A′ or Aᶜ)

Given a Universal Set U, A′ = {x ∈ U : x ∉ A}

Example 4: If U = {1, 2, 3, 4, 5, 6, 7, 8} and A = {2, 4, 6, 8}, then A′ = {1, 3, 5, 7} ✓

Venn Diagrams — Visualising Set Operations

How to Draw a Venn Diagram

Draw a RECTANGLE to represent the UNIVERSAL SET (U).
Draw CIRCLES inside for each set.
Shade the RELEVANT region for the operation.

Venn Diagrams for Common Operations

A ∪ B: Shade ALL regions inside both circles (including overlap). A ∩ B: Shade ONLY the OVERLAPPING region between the two circles. A − B: Shade the part of circle A that does NOT overlap with circle B. A′: Shade everything OUTSIDE circle A (but inside U).

'For THREE sets, draw three overlapping circles. The centre belongs to ALL three sets.'

Example with Three Sets

If U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 2, 3, 4, 5}, B = {3, 4, 5, 6, 7}, C = {5, 7, 8, 9}:

n(A) = 5, n(B) = 5, n(C) = 4 n(A ∩ B) = 3, n(B ∩ C) = 2, n(A ∩ C) = 1, n(A ∩ B ∩ C) = 1

Cardinality of Sets

The CARDINAL NUMBER n(A) is the NUMBER of elements in set A.

Cardinality Formulas

Formula	Meaning
n(A ∪ B) = n(A) + n(B) − n(A ∩ B)	Two sets — SUBTRACT overlap once
n(U) = n(A) + n(A′)	Universal set = set + its complement
n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A∩B) − n(B∩C) − n(A∩C) + n(A∩B∩C)	Three sets — inclusion-exclusion principle

'For two sets, subtract the intersection ONCE. For three sets, subtract PAIRWISE intersections, then ADD BACK the triple intersection.'

Worked Example — Two Sets

In a class of 60 students, 35 like Mathematics, 28 like Science, and 15 like BOTH. Find how many like: a) At least one subject b) Neither subject c) Only Mathematics d) Only Science

n(M) = 35, n(S) = 28, n(M ∩ S) = 15

a) n(M ∪ S) = n(M) + n(S) − n(M ∩ S) = 35 + 28 − 15 = 48 ✓ b) n(Neither) = n(U) − n(M ∪ S) = 60 − 48 = 12 ✓ c) Only M = n(M) − n(M ∩ S) = 35 − 15 = 20 ✓ d) Only S = n(S) − n(M ∩ S) = 28 − 15 = 13 ✓

'Always draw the Venn diagram first. It helps you SEE the numbers before you calculate.'

Worked Example — Three Sets

In a survey of 100 families, 55 read Times of India, 40 read Hindu, 35 read Deccan Chronicle. 20 read TOI and Hindu, 15 read Hindu and DC, 10 read TOI and DC, and 5 read ALL three. Find: a) Families reading at least one newspaper b) Families reading exactly one newspaper

Let T = TOI, H = Hindu, D = Deccan Chronicle

n(T ∪ H ∪ D) = n(T) + n(H) + n(D) − n(T∩H) − n(H∩D) − n(T∩D) + n(T∩H∩D) = 55 + 40 + 35 − 20 − 15 − 10 + 5 = 90 ✓

90 families read at least one newspaper.

For exactly one: Only T = n(T) − n(T∩H) − n(T∩D) + n(T∩H∩D) = 55 − 20 − 10 + 5 = 30 Only H = n(H) − n(T∩H) − n(H∩D) + n(T∩H∩D) = 40 − 20 − 15 + 5 = 10 Only D = n(D) − n(H∩D) − n(T∩D) + n(T∩H∩D) = 35 − 15 − 10 + 5 = 15 Exactly one = 30 + 10 + 15 = 55 ✓

De Morgan's Laws

Law 1: (A ∪ B)′ = A′ ∩ B′ 'The complement of a UNION is the INTERSECTION of the complements.' Law 2: (A ∩ B)′ = A′ ∪ B′ 'The complement of an INTERSECTION is the UNION of the complements.'

Verification with Example

Let U = {1, 2, 3, 4, 5, 6, 7, 8}, A = {2, 4, 6}, B = {1, 2, 3, 4}

A ∪ B = {1, 2, 3, 4, 6} → (A ∪ B)′ = {5, 7, 8} A′ = {1, 3, 5, 7, 8}, B′ = {5, 6, 7, 8} → A′ ∩ B′ = {5, 7, 8} ✓ Hence (A ∪ B)′ = A′ ∩ B′ — De Morgan's 1st Law VERIFIED.

A ∩ B = {2, 4} → (A ∩ B)′ = {1, 3, 5, 6, 7, 8} A′ ∪ B′ = {1, 3, 5, 7, 8} ∪ {5, 6, 7, 8} = {1, 3, 5, 6, 7, 8} ✓ Hence (A ∩ B)′ = A′ ∪ B′ — De Morgan's 2nd Law VERIFIED.

Types of Sets — Detailed

Type	Definition	Example
Empty (Null) Set	No elements. Symbol: ∅ or {}	{ } or ∅
Singleton Set	Exactly ONE element	{5}
Finite Set	Countable number of elements	{a, b, c}
Infinite Set	Countless elements	ℕ = {1, 2, 3, ...}
Equal Sets	Identical elements (order irrelevant)	{1, 2} = {2, 1}
Equivalent Sets	SAME number of elements	A = {p, q}, B = {2, 4} → n(A) = n(B) = 2
Disjoint Sets	NO common elements. A ∩ B = ∅	A = {1, 2}, B = {3, 4}
Power Set	Set of ALL subsets. n[P(A)] = 2ⁿ	A = {a, b}, P(A) = {∅, {a}, {b}, {a, b}}

'Power set includes the EMPTY SET and the SET ITSELF. For a set with n elements, the power set has 2ⁿ elements.'

Subsets — Deep Dive

A ⊆ B means EVERY element of A is in B.

Proper Subset (A ⊂ B): A ⊆ B BUT A ≠ B (at least one element of B is missing in A).

Number of Subsets of a set with n elements:

Total subsets: 2ⁿ
Proper subsets: 2ⁿ − 1 (exclude the set itself)
Non-empty subsets: 2ⁿ − 1 (exclude empty set)
Non-empty proper subsets: 2ⁿ − 2

Example: For A = {x, y, z}, n = 3 Total subsets = 2³ = 8: ∅, {x}, {y}, {z}, {x,y}, {x,z}, {y,z}, {x,y,z} Proper subsets = 8 − 1 = 7 (exclude {x,y,z}) Non-empty subsets = 8 − 1 = 7 (exclude ∅)

Common Mistakes

'Set {0} is empty' — No! {0} has ONE element (0). The empty set is {} or ∅.
'Order matters in sets' — NO! {1, 2, 3} = {3, 1, 2}. Sets care about MEMBERSHIP, not ORDER.
'A − B = B − A' — FALSE. Set difference is NOT commutative.
'n(A ∪ B) = n(A) + n(B)' — Wrong unless A and B are DISJOINT. Always subtract n(A ∩ B).
'∅ and {∅} are the same' — Different. ∅ has 0 elements. {∅} has 1 element (the empty set).

AP SSC Board Exam Focus

Topic	Marks	Frequency
Set operations (∪, ∩, −, ′)	2-3	Very High
Cardinality problems (2 sets)	4	Very High
Venn diagram interpretation	3-4	High
De Morgan's laws	2	Moderate
Types of sets	1-2	Moderate
Power set and subsets	2	Moderate
Cardinality problems (3 sets)	4	High

Self-Test Questions

If A = {x : x is a prime number less than 15} and B = {x : x is an odd number less than 15}, find A ∪ B, A ∩ B, A − B, and B − A.
In a school of 500 students, 280 play cricket, 180 play football, and 50 play BOTH. How many play NEITHER sport?
In a group of 200 people, 120 like tea, 110 like coffee, and 30 like NEITHER. How many like BOTH tea and coffee?
If U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {2, 4, 6, 8, 10}, B = {1, 3, 5, 7, 9}, verify De Morgan's laws.
For set A = {a, b, c, d}, write the power set and find the number of proper subsets.
In a class, 50 students took Hindi, 40 took English, 30 took BOTH. If the class has 80 students, how many took NEITHER?
If n(A) = 25, n(B) = 30, n(A ∪ B) = 45, find n(A ∩ B).
Prove that A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) using set membership.

Answers: 1) A = {2,3,5,7,11,13}, B = {1,3,5,7,9,11,13}, A∪B={1,2,3,5,7,9,11,13}, A∩B={3,5,7,11,13}, A−B={2}, B−A={1,9}, 2) 90, 3) 60, 4) Verified above, 5) 16 subsets, 15 proper, 6) 20, 7) 10, 8) Use membership table

Key formulas & results

Everything you need to memorise, in one card. Screenshot this for revision.

Set Operations and Cardinal Number Formula

UNION (A∪B): All elements in A OR B (or both). INTERSECTION (A∩B): Only elements in BOTH A and B. DIFFERENCE (A−B): Elements in A but NOT in B. COMPLEMENT (A'): All elements in universal set U but not in A. CARDINAL NUMBER FORMULA: n(A∪B) = n(A) + n(B) − n(A∩B). POWER SET: Set of all subsets of A. If n(A)=k, then Power set has 2ᵏ elements. NUMBER OF SUBSETS: A set with n elements has 2ⁿ subsets. Proper subsets: 2ⁿ−1.

WORD PROBLEM WITH CARDINAL FORMULA: 'In a class of 40 students, 25 play cricket, 20 play football, and 10 play both. How many play at least one game? How many play neither?' SOLUTION: n(C∪F) = n(C)+n(F)−n(C∩F) = 25+20−10 = 35. Play at least one = 35. Neither = 40−35 = 5. AP SSC uses this formula in survey problems.

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Common mistakes & fixes

These are the exact errors that cost students marks in board exams. Read them once, save yourself the trouble.

WATCH OUT

✗ Confusing subset with proper subset, and equal sets with equivalent sets

✓ EQUAL SETS: A = B if they have the SAME elements (order doesn't matter, repetitions don't count). {1,2,3} = {3,1,2} = {1,1,2,3}. EQUIVALENT SETS: n(A) = n(B) — same NUMBER of elements. {1,2,3} and {a,b,c} are equivalent but not equal. SUBSET: A ⊆ B — every element of A is in B. Every set is a subset of itself. PROPER SUBSET: A ⊂ B — A ⊆ B AND A ≠ B (A is 'strictly inside' B, not equal).

Practice problems

Work through this chapter's problems as a readiness check — reveal each solution, mark yourself honestly, and get your gap report at the end.

Readiness check

Are you exam-ready for Sets?

1 problems from this chapter. Try each one, reveal the worked solution, mark yourself honestly — get your gap report at the end.

1 questions~2 min

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•SET NOTATION: ∈ (belongs to), ∉ (doesn't belong), ⊆ (subset), ⊂ (proper subset), ∅ (empty set), n(A) (cardinality = number of elements). Curly braces {} enclose set elements. A = {1,2,3} and B = {3,2,1} → A = B (order doesn't matter in sets).
•SET TYPES: Empty (∅, no elements). Finite (countable). Infinite (N, Z, R). Equal (same elements). Equivalent (same cardinality, n(A)=n(B)). Universal (U) = all elements under consideration. Subset: every element of A is in B. Proper subset: A ⊂ B means A ⊆ B AND A ≠ B.
•SET OPERATIONS: Union (A∪B) = all elements in A OR B. Intersection (A∩B) = elements in BOTH A and B. Difference (A−B) = elements in A but NOT in B. Complement (A') = all elements in U but not in A. De Morgan's: (A∪B)' = A'∩B' and (A∩B)' = A'∪B'.
•CARDINAL NUMBER FORMULA: n(A∪B) = n(A) + n(B) − n(A∩B). 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). These are the survey/classification word problem formulas.
•SUBSETS COUNT: A set with n elements has 2ⁿ subsets (including empty set and the set itself). Proper subsets = 2ⁿ − 1. Power set P(A) = collection of all subsets of A. n(P(A)) = 2ⁿ.
•VENN DIAGRAMS: Two overlapping circles inside a rectangle (rectangle = Universal set U). Region inside only A = A−B. Region inside only B = B−A. Overlapping region = A∩B. All inside either circle = A∪B. Outside both circles = (A∪B)' = A'∩B'.
•WORD PROBLEM APPROACH: (1) Define sets clearly (let M = set of people who like cricket). (2) Identify what n(M), n(F), n(M∩F), n(M∪F), n(U) are. (3) Apply the cardinal formula. (4) 'Neither' = n(U) − n(M∪F). 'Only M' = n(M) − n(M∩F). 'At least one' = n(M∪F).
•DISJOINT SETS: A and B are disjoint if A∩B = ∅ (no common elements). If disjoint, n(A∪B) = n(A) + n(B) (no subtraction needed). Example: A = {odd numbers}, B = {even numbers} — disjoint.
•FINITE SET vs INFINITE SET: Natural numbers N = {1,2,3,...} — infinite. Set of CBSE students in India — finite (large but countable). Set of real numbers between 0 and 1 — infinite. The empty set has 0 elements and is FINITE.
•AP BOARD KEY: The n(A∪B) formula is the most tested. Always check if the problem gives n(A∩B) directly or if you need to find it from n(A∪B). Common trap: 'How many belong to exactly one set' = n(A) + n(B) − 2×n(A∩B) (NOT n(A∪B) − n(A∩B)). Drawing a Venn diagram first always prevents this error.

Andhra Pradesh (BIEAP) marks blueprint

Where the marks come from in this chapter — so you can plan your prep.

Where this shows up in the real world

This chapter isn't just an exam topic — it lives in the world around you.

Database queries and SQL

Every database query uses set operations. SQL's JOIN operation is essentially A∩B (finding records that exist in both tables). UNION combines two result sets (A∪B). WHERE NOT IN is set difference (A−B). Every time an app retrieves data — finding customers who bought product X AND product Y — it is computing an intersection. The set theory from Class 10 is the mathematical foundation of all database logic.

Survey analysis and market research

The cardinal number formula is used in every survey that asks about overlapping categories. 'How many customers use both our mobile app and website?' requires n(A∩B) = n(A) + n(B) − n(A∪B). Market research firms use Venn diagram analysis to understand audience overlap between different media channels, products, or demographics. The Class 10 formula directly applies.

Search engine boolean queries

When you type a search query like 'cricket AND India' — the search engine finds documents that contain 'cricket' AND 'India' — an intersection operation. 'cricket OR football' is a union. 'cricket NOT football' is set difference. The entire logic of information retrieval in Google, Bing, and academic databases is built on Boolean set algebra — the same operations studied in this chapter.

Exam strategy

Battle-tested tips from teachers and toppers for this chapter.

Cardinal formula word problems (2 marks): write the formula first — n(A∪B) = n(A) + n(B) − n(A∩B). Then substitute given values. Find the unknown. A drawn Venn diagram (even small) shows examiner you understand the structure and earns partial marks.
Set operations (2 marks): to find A∪B, A∩B, A−B, B−A from given sets — list ALL elements first, then apply the operation definition. Don't rush. Check: n(A∪B) = n(A−B) + n(B−A) + n(A∩B) is a good self-check.
Subsets counting: if asked 'number of subsets of a set with n elements', the answer is 2ⁿ. If asked 'proper subsets', it is 2ⁿ − 1. Write the formula and substitute — don't list all subsets.
Venn diagrams: draw clearly, label all regions. The four regions are: A only, B only, A∩B, outside both. Fill in numbers from INSIDE OUT: fill A∩B first, then A-only, then B-only, then outside.
Equal vs equivalent: if asked 'are these sets equal or equivalent?', check elements for equality first; if not equal, check if cardinalities are the same (equivalent). Write both steps explicitly.

Going beyond the textbook

For olympiad aspirants and curious learners — topics that build on this chapter.

Research Georg Cantor (1845–1918) — the mathematician who invented set theory. Cantor proved that some infinities are larger than others: the set of natural numbers (N) and the set of real numbers (R) are both infinite, but R is 'more infinite' — Cantor called these different cardinalities ℵ₀ (aleph-null, the 'smallest' infinity) and c (the continuum). His diagonal argument proving that real numbers are uncountable is one of the most elegant proofs in mathematics. Research it.
Investigate Russell's Paradox (Bertrand Russell, 1901) — a paradox in set theory: 'Consider the set R of all sets that do not contain themselves. Does R contain itself?' If yes → R should not contain itself (contradiction). If no → R should contain itself (contradiction). This paradox showed that 'the set of all sets' cannot exist and led to the development of Axiomatic Set Theory (ZFC — Zermelo-Fraenkel axioms). It forced mathematicians to rethink the foundations of all mathematics.
Explore Venn diagrams beyond two or three sets. For four sets, Venn diagrams become impossible with just circles — you need ellipses or other shapes. Researchers have studied the symmetry conditions under which n-set Venn diagrams can exist. For n=5, a symmetric Venn diagram requires exactly 5 congruent curved regions — the first was discovered in 1964. Research the problem of creating Venn diagrams for large n.
Research how sets underlie all of modern mathematics: functions are defined as sets of ordered pairs, relations are subsets of cartesian products, numbers themselves can be defined as sets (von Neumann ordinals: 0=∅, 1={∅}, 2={∅,{∅}}, etc.). The entire edifice of calculus, algebra, and number theory ultimately rests on set theory. Explore the Peano axioms that define natural numbers from set theory.

Where else this chapter is tested

CBSE board isn't the only one — other exams test this chapter too.

AP Board SSC (Class 10)Medium — Sets appears for 2–4 marks in AP SSC Mathematics; the cardinal formula word problem is the most frequently tested question type

JEE Main and AdvancedHigh — Set theory, relations, and functions are among the most tested topics in JEE Mathematics; Class 10 set operations are the foundation

NTSE (Mathematics section)Medium — Set operations and Venn diagrams appear in NTSE Stage I mathematics

AP EAPCET (Engineering entrance)High — Sets, relations, and functions form a full chapter in EAPCET Mathematics; mastering Class 10 sets is the necessary first step

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Questions students ask

The real ones — pulled from the Q&A community and tutor sessions.

EQUAL SETS: Two sets A and B are equal (A = B) if and only if they have exactly the SAME ELEMENTS — regardless of order or repetition. {1,2,3} = {3,2,1} = {1,2,2,3}. Every element of A is in B AND every element of B is in A. EQUIVALENT SETS: Two sets A and B are equivalent if they have the SAME NUMBER OF ELEMENTS (same cardinality) — the elements themselves may be completely different. {1,2,3} and {a,b,c} are equivalent (n = 3 each) but not equal. Note: All equal sets are equivalent, but not all equivalent sets are equal.

The full formula n(A∪B) = n(A) + n(B) − n(A∩B) is ALWAYS correct. When we add n(A) + n(B), we count the elements in A∩B TWICE (once for A, once for B). We subtract n(A∩B) once to correct this double-counting. SPECIAL CASE: If A and B are DISJOINT (A∩B = ∅, n(A∩B) = 0), the formula becomes n(A∪B) = n(A) + n(B) — no subtraction needed. For AP SSC word problems, always use the full formula unless the problem explicitly says the sets are disjoint.

A set with n elements has 2ⁿ subsets. For n=4: 2⁴ = 16 subsets. This includes the empty set (∅) and the set itself. Proper subsets = 2⁴ − 1 = 15 (excluding the set itself). Why 2ⁿ? For each element, there are two choices: include it in the subset or not. With 4 elements, there are 2×2×2×2 = 16 possible combinations. Example: {a,b} has 4 subsets: ∅, {a}, {b}, {a,b}.

COMPLEMENT (A'): ALL elements in the Universal set U that are NOT in A. It depends on what U is defined to be. Example: If U = {1,2,3,4,5,6,7,8,9,10} and A = {1,2,3,4,5}, then A' = {6,7,8,9,10}. SET DIFFERENCE (A−B): Elements in A that are NOT in B. It doesn't involve U. Example: A = {1,2,3,4,5}, B = {3,4,5,6,7}. A−B = {1,2} (elements in A but not in B). KEY DISTINCTION: Complement always uses U; set difference does not. A' = U − A is the special case of set difference with U as the first set.

This is a standard cardinal formula word problem. n(T∪C) = n(T) + n(C) − n(T∩C) = 60 + 50 − 30 = 80. People who like at least one = 80. People who like NEITHER = Total − n(T∪C) = 100 − 80 = 20. Drawing the Venn diagram confirms this: only tea = 60−30 = 30; only coffee = 50−30 = 20; both = 30; total in circles = 30+30+20 = 80; outside = 100−80 = 20 ✓.

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