These sets class 11 notes are a quick revision guide for Class 11 Maths Chapter 1 Sets, built from the NCERT textbook. A set is the building block of mathematics — the chapters on relations, functions, sequences and probability all speak this language.
The page gives you definitions and notations in one table, the laws of union and intersection, De Morgan’s laws, four fully solved examples with fresh numbers, the mistakes that lose marks, and a one-page recap for the night before the exam. Use the table of contents to jump to any section.
Sets Class 11 Notes: Chapter Overview
Chapter 1 is the opening chapter of your Class 11 Mathematics course. It builds the vocabulary of the subject: what a set is, how to write one, the different kinds of sets, and the operations on them — union, intersection, difference and complement. It also introduces Venn diagrams as a visual tool and closes with De Morgan’s laws.
Practice in the chapter is organised into five exercises (1.1 to 1.5) plus a Miscellaneous exercise, across about 23 pages of the textbook. The chapter carries roughly 2 marks in the board exam, but it quietly matters more: every later chapter reuses its notation. The official chapter text is available as a PDF from NCERT if you want the full worked examples.
Key Concepts: Sets, Representations & Special Sets
What exactly is a set?
A set is a well-defined collection of objects (NCERT, p. 2). “Well-defined” means you can always decide, for any given object, whether it belongs to the collection or not.
- Well-defined: vowels in the English alphabet, natural numbers less than 10, prime factors of 210, solutions of \( x^2 – 5x + 6 = 0 \).
- Not well-defined: “ten most talented writers of India” — the judgement differs from person to person.
The objects inside a set are called its elements or members. Sets are named with capital letters (A, B, U); elements with small letters. The symbol ∈ means “belongs to”, so \( a \in A \) reads “a is an element of A”. Its negation is ∉ (NCERT, p. 2).
Roster form vs set-builder form
Every set can be written in two ways (NCERT, p. 2). In roster (tabular) form you list the elements inside braces. In set-builder form you describe the single property shared by all elements. The colon “:” means “such that”. Here is the comparison table with original examples:
| Feature | Roster (tabular) form | Set-builder form |
|---|---|---|
| Form | All elements listed in braces | A defining property in braces |
| Best for | Small, finite sets | Large or infinite sets |
| Example | \( \{7, 14, 21, 28\} \) | \( \{x : x = 7n,\ n \in \mathbb{N},\ n \le 4\} \) |
| Read as | “elements 7, 14, 21, 28” | “all x such that x is a multiple of 7 up to 28” |
| Watch out | Order is irrelevant; never repeat an element | Colon “:” stands for “such that” |
So \( \{7, 14, 21, 28\} \) and \( \{x : x = 7n,\ n \in \mathbb{N},\ n \le 4\} \) are the same set written in the two forms.
Standard number sets
These special sets appear throughout the chapter and in the whole course (NCERT, p. 2 and p. 10):
| Symbol | Name | Contents |
|---|---|---|
| \( \mathbb{N} \) | Natural numbers | \( \{1, 2, 3, 4, 5, \dots\} \) |
| \( \mathbb{Z} \) | Integers | \( \{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\} \) |
| \( \mathbb{Q} \) | Rational numbers | \( \{x : x = \frac{p}{q}, p, q \in \mathbb{Z}, q \neq 0\} \) |
| \( \mathbb{T} \) | Irrational numbers | \( \{x : x \in \mathbb{R},\ x \notin \mathbb{Q}\} \) e.g. \( \sqrt{2}, \pi \) |
| \( \mathbb{R} \) | Real numbers | All rational and irrational numbers |
| \( \mathbb{Z}^+, \mathbb{Q}^+, \mathbb{R}^+ \) | Positive parts | Positive integers, rationals, reals |
The relations among them: \( \mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \), \( \mathbb{T} \subset \mathbb{R} \) and \( \mathbb{N} \not\subset \mathbb{T} \) (NCERT, p. 11).
Intervals: subsets of \( \mathbb{R} \)
An interval is a set of real numbers lying between two endpoints (NCERT, p. 11). Whether the endpoints are included decides the brackets:
| Set-builder | Interval | Say it aloud | Venn / number-line picture |
|---|---|---|---|
| \( \{x : a \lt x \lt b\} \) | \( (a, b) \) open | “a to b, both excluded” | Open circles at a and b — the line between them is included |
| \( \{x : a \le x \le b\} \) | \( [a, b] \) closed | “a to b, both included” | Filled dots at a and b — whole segment shaded |
| \( \{x : a \le x \lt b\} \) | \( [a, b) \) half-open | “a included, b excluded” | Filled dot at a, open circle at b |
| \( \{x : a \lt x \le b\} \) | \( (a, b] \) half-open | “a excluded, b included” | Open circle at a, filled dot at b |
The length of any interval is \( b – a \) (NCERT, p. 12) — it is the same for all four forms. Example: \( \{x : x \in \mathbb{R},\ -5 \lt x \le 7\} \) is the interval \( (-5, 7] \), and its length is \( 7 – (-5) = 12 \). In a Venn diagram, the universal set \( \mathbb{R} \) is the rectangle and an interval is a segment of the number line inside it.
Real-life use: search filters and databases
Sets are not abstract. On a shopping site, ticking “under ₹500” and “free delivery” shows you the intersection of two sets of products. A database query such as customers in Delhi AND who bought in March is an intersection; Delhi OR March is a union; NOT in Delhi is the complement with the universe being all customers.
Every filter bar you use runs on set operations.
Definitions and Notations at a Glance
This table carries every term you must know, with its symbol and an example.
| Term | Meaning | Example / Symbol |
|---|---|---|
| Set | Well-defined collection of objects (p. 2) | \( \{a, e, i, o, u\} \) |
| Element / member | An object inside a set (p. 2) | \( a \in V \) |
| Roster form | Elements listed in braces (p. 2) | \( \{2, 4, 6\} \) |
| Set-builder form | Elements described by a property (p. 3) | \( \{x : x \text{ is even},\ x \lt 8\} \) |
| Empty / null set | Set with no elements (p. 5) | \( \phi \) or \( \{\} \) |
| Finite set | Contains a definite number of elements (p. 6) | The days of the week |
| Infinite set | Not finite (p. 6) | \( \mathbb{N} = \{1, 2, 3, \dots\} \) |
| Equal sets \( A = B \) | Exactly the same elements (p. 7) | \( \{1, 2, 3\} = \{3, 2, 1\} \) |
| Subset \( A \subset B \) | Every element of A is in B (p. 9) | \( \{a\} \subset \{a, b, c\} \) |
| Proper subset | \( A \subset B \) and \( A \neq B \) (p. 9) | \( \{1, 2\} \subset \{1, 2, 3\} \) |
| Superset | B is a superset of A when \( A \subset B \) (p. 9) | \( \{1, 2, 3\} \supset \{1, 2\} \) |
| Singleton set | A set with exactly one element (p. 9) | \( \{5\} \) |
| Power set \( P(A) \) | The set of all subsets of A | For \( A = \{a, b\} \), \( P(A) = \{\phi, \{a\}, \{b\}, \{a, b\}\} \) |
| Universal set \( U \) | The basic set for a context (p. 12) | When studying integers, a possible U is \( \mathbb{R} \) |
Two rules to lock in now: the empty set is a subset of every set, and every set is a subset of itself, \( A \subset A \) (NCERT, p. 9).
Important Properties and Formula Summary
Laws of union and intersection
| Law | Union | Intersection |
|---|---|---|
| Commutative (p. 14, 15) | \( A \cup B = B \cup A \) | \( A \cap B = B \cap A \) |
| Associative (p. 14, 15) | \( (A \cup B) \cup C = A \cup (B \cup C) \) | \( (A \cap B) \cap C = A \cap (B \cap C) \) |
| Idempotent (p. 14, 15) | \( A \cup A = A \) | \( A \cap A = A \) |
| Identity (p. 14, 15) | \( A \cup \phi = A \) | \( A \cap U = A \) |
| Domination / law of U and φ (p. 14, 15) | \( A \cup U = U \) | \( A \cap \phi = \phi \) |
| Distributive (p. 15) | — | \( A \cap (B \cup C) = (A \cap B) \cup (A \cap C) \) |
Why they hold: commutative and associative laws work because “or” and “and” are symmetric ideas — joining A with B is the same as joining B with A. The distributive law is the set version of “and” spreading over “or”, mirroring how \( a \times (b + c) = ab + ac \) spreads in arithmetic.
Complement laws (NCERT, p. 20)
- Complement law: \( A \cup A’ = U \) and \( A \cap A’ = \phi \).
- Law of double complement: \( (A’)’ = A \).
- Laws of U and φ: \( \phi’ = U \) and \( U’ = \phi \).
- Definition: \( A’ = U – A = \{x : x \in U,\ x \notin A\} \) (NCERT, p. 18).
- De Morgan’s laws (NCERT, p. 19-20): \( (A \cup B)’ = A’ \cap B’ \) and \( (A \cap B)’ = A’ \cup B’ \).
De Morgan’s laws in words: the complement of a union is the intersection of the complements, and the complement of an intersection is the union of the complements.
Worked Examples: Step-by-Step Problems
These four problems use fresh numbers so you can practise the method, not memorise an answer.
Worked Example 1: Converting between roster and set-builder form
- Step 1: Write \( B = \{x : x = 3n,\ n \in \mathbb{N},\ 1 \le n \le 4\} \) in roster form.
- Step 2: Put \( n = 1, 2, 3, 4 \) into \( 3n \): we get 3, 6, 9, 12.
\[ B = \{3, 6, 9, 12\} \]
Step 3: Now write \( C = \{-2, -1, 0, 1, 2, 3\} \) in set-builder form.
The common property is “integer from −2 to 3”.
\[ C = \{x : x \in \mathbb{Z},\ -2 \le x \le 3\} \]
Final answer: \( B = \{3, 6, 9, 12\} \) and \( C = \{x : x \in \mathbb{Z},\ -2 \le x \le 3\} \).
Worked Example 2: Checking whether two sets are equal
Step 1: A set does not change if elements are repeated or listed in a different order (NCERT, p. 7).
Let \( A = \{x : x \text{ is a letter of the word MAMMA}\} \) and \( B = \{M, A\} \).
Step 2: The distinct letters of MAMMA are M and A, so \( A = \{M, A\} \).
\[ A = \{M, A\} = B \]
Step 3: Check a non-equal pair.
Let \( P = \{2, 3, 5, 7\} \) (primes under 10) and \( Q = \{x : x \text{ divides } 30\} = \{1, 2, 3, 5, 6, 10, 15, 30\} \).
Since 1, 6, 10, 15, 30 are in Q but not in P, the sets differ.
Final answer: \( A = B \) (elements repeated and reordered make no difference), but \( P \neq Q \) because Q has extra elements.
Worked Example 3: Union, intersection and difference
- Step 1: Take \( A = \{2, 4, 6, 8, 10\} \) and \( B = \{4, 8, 12, 16\} \).
- Step 2: Union takes every element of A and every element of B, common elements once.
\[ A \cup B = \{2, 4, 6, 8, 10, 12, 16\} \]
Step 3: Intersection keeps only common elements.
\[ A \cap B = \{4, 8\} \]
Step 4: Difference removes from A everything that is in B.
\[ A – B = \{2, 6, 10\}, \quad B – A = \{12, 16\} \]
Final answer: \( A \cup B = \{2, 4, 6, 8, 10, 12, 16\} \), \( A \cap B = \{4, 8\} \), \( A – B = \{2, 6, 10\} \). Note \( A – B \neq B – A \) — set difference is not commutative.
Worked Example 4: Verifying De Morgan’s law step by step
- Step 1: Let \( U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \), \( A = \{3, 5, 7, 9\} \), \( B = \{2, 3, 5, 7\} \).
- Step 2: First law \( (A \cup B)’ = A’ \cap B’ \).
Form the union, then take its complement.
\[ A \cup B = \{2, 3, 5, 7, 9\}, \quad (A \cup B)’ = \{1, 4, 6, 8, 10\} \]
Step 3: Now compute the right side separately.
\[ A’ = \{1, 2, 4, 6, 8, 10\}, \quad B’ = \{1, 4, 6, 8, 9, 10\} \]
\[ A’ \cap B’ = \{1, 4, 6, 8, 10\} \]
- Step 1: Both sides equal \( \{1, 4, 6, 8, 10\} \), so the first law holds.
- Step 2: Second law \( (A \cap B)’ = A’ \cup B’ \).
\[ A \cap B = \{3, 5, 7\}, \quad (A \cap B)’ = \{1, 2, 4, 6, 8, 9, 10\} \]
\[ A’ \cup B’ = \{1, 2, 4, 6, 8, 9, 10\} \]
Final answer: \( (A \cup B)’ = A’ \cap B’ = \{1, 4, 6, 8, 10\} \) and \( (A \cap B)’ = A’ \cup B’ = \{1, 2, 4, 6, 8, 9, 10\} \) — both De Morgan laws verified. This is exactly how a 4-mark verification question should be laid out: compute each side, then state the equality.
Common Mistakes and How to Avoid Them
The table below is the chapter’s error-correction autopsy — the mistakes that actually appear in exam scripts, with the rule and a quick self-check.
| Mistake | Correct rule | How to check your answer |
|---|---|---|
| Writing an element twice in roster form: \( \{1, 1, 2, 3\} \) | Repetition is allowed but pointless — a set lists each distinct element once (NCERT, p. 3) | Delete every duplicate; the set must not change |
| Mixing up ∈ and ⊂: “\( \{a\} \in \{a, b, c\} \)” or “\( 1 \subset A \)” | ∈ joins an element to a set; ⊂ joins a set to a set (NCERT, p. 9) | Ask: is the left object a single item or a collection? \( a \in A \), but \( \{a\} \subset A \) |
| Forgetting the empty set is a subset of every set | \( \phi \subset A \) for every set A (NCERT, p. 9) | In any subset listing, always start with \( \phi \) |
| Writing \( \{x : 1 \le x \le 5\} \) as an open interval | \( \le \) means the endpoint is included → closed bracket (NCERT, p. 11) | Each \( \le \) maps to a square bracket; each \( \lt \) maps to a round bracket |
| Misapplying De Morgan: writing \( (A \cup B)’ = A’ \cup B’ \) | The “not” distributes and flips the operation: \( (A \cup B)’ = A’ \cap B’ \) (NCERT, p. 19) | Test on a small U — the wrong version never matches |
| Error-correction autopsy — confusing \( \neq \) with \( \not\subset \) | \( A \neq B \) means “the two sets differ”; \( A \not\subset B \) means “A is not a subset of B” — different statements | Example: \( A = \{3, 9\}, B = \{3, 9, 12\} \). Here \( A \neq B \) is true, but \( A \not\subset B \) is false since A is a subset of B. Read the letters carefully. |
One more trap from the chapter: an element of a set can never be a subset of itself. In \( A = \{1, 2, \{3, 4\}, 5\} \), the element \( \{3, 4\} \) satisfies \( \{3, 4\} \in A \) and \( \{\{3, 4\}\} \subset A \), but \( \{3, 4\} \not\subset A \) (NCERT, p. 12).
Exam Pointers: What to Expect and How to Prepare
This chapter contributes about 2 marks to the board exam, and the questions are highly predictable in form:
- 1-mark questions: say whether a collection is a set; fill ∈ or ∉; write a small set in roster or set-builder form; identify the empty set.
- 3-mark questions: show two sets are equal; list all subsets of a small set like \( \{a, b, c\} \); convert between intervals and set-builder form.
- 4-mark questions: perform union, intersection, difference and complement together; verify De Morgan’s laws for given sets.
Examiners reward accurate notation more than anything here — a missing bracket or a \( \subset \) where ∈ belongs costs the mark even if the logic is right. Write every step: state the universal set first, then each intermediate set, then the final equality.
Quick Revision Recap
One scan of this table before the paper:
| Concept | Key point | Notes |
|---|---|---|
| Set | Well-defined collection of objects | Must be decidable for every object |
| Empty set | No elements: \( \phi \) or \( \{\} \) | Subset of every set |
| Finite / infinite | Definite count vs not finite | \( \mathbb{N} \) is infinite |
| Equal sets | Exactly the same elements | Order and repetition ignored |
| Subset | \( A \subset B \): every element of A in B | Proper subset: \( A \neq B \) too |
| Intervals | \( (a,b), [a,b], [a,b), (a,b] \) | Length always \( b – a \) |
| Union / intersection | \( \cup \) joins, \( \cap \) keeps common | Idempotent, commutative, associative |
| Difference | \( A – B \): in A, not in B | \( A – B \neq B – A \) in general |
| Complement | \( A’ = U – A \) | \( (A’)’ = A \) |
| De Morgan | \( (A \cup B)’ = A’ \cap B’ \), \( (A \cap B)’ = A’ \cup B’ \) | Complement flips the operation |
Frequently Asked Questions
What is the empty set and when do we use it?
The empty set (also called the null or void set) contains no elements and is written \( \phi \) or \( \{\} \) (NCERT, p. 5). Example: \( \{x : x \text{ is a natural number and } 1 \lt x \lt 2\} = \phi \) because no natural number lies strictly between 1 and 2.
You use it when a condition is satisfied by nothing at all, and when two sets are disjoint — that is, their intersection is the empty set, \( A \cap B = \phi \).
How do I convert roster form to set-builder form?
Look for the single property every element shares. For \( \{4, 9, 16, 25\} \), each element is a square of a natural number from 2 to 5, so write \( \{x : x = n^2,\ n \in \mathbb{N},\ 2 \le n \le 5\} \). To go the other way, substitute the allowed values and list the results.
What is the difference between subset and proper subset?
\( A \subset B \) means every element of A is in B — A may equal B. A proper subset adds the condition \( A \neq B \), so B contains at least one element not in A. For \( A = \{1\} \) and \( B = \{1\} \), we have \( A \subset B \) but A is not a proper subset of B; for \( B = \{1, 2\} \), A is a proper subset.
How do I verify De Morgan’s law for two sets?
Fix a universal set U. Compute \( A \cup B \), then take its complement; separately compute \( A’ \cap B’ \). If both sides match, the law holds for those sets — see Worked Example 4 above, which does exactly this for \( U = \{1, 2, \dots, 10\} \). Repeat for the second law \( (A \cap B)’ = A’ \cup B’ \).
What is the length of an interval like (3, 8]?
The length of any interval is \( b – a \) where a and b are the endpoints, regardless of which brackets are used (NCERT, p. 12). So the length of \( (3, 8] \) is \( 8 – 3 = 5 \). The bracket style changes which points belong to the set, not the distance between them.
If you need the full chapter view, these Relations and Functions notes continue straight from sets — relations are just sets of ordered pairs. Browse the Class 11 revision notes and the wider CBSE notes library for the other chapters.
Reference: NCERT Class 11 Mathematics textbook, chapter SETS.
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