Sets Class 11 Notes: Types, Operations & Venn Diagrams

These sets class 11 notes cover the first chapter of NCERT Class 11 Maths. A set is a well-defined collection of objects, and the objects are called its elements. Sets form the language of the whole of modern mathematics, so the chapter also supports relations, functions and probability later. First, learn the basic terms. Next, study the types of sets and the operations. Finally, practise Venn diagrams and word problems.

What Is a Set? Basic Terms

A set must be well-defined. For example, “the vowels in the English alphabet” is a set, but “the best players in a school” is not. The reason is that opinions differ about “best”.

We name sets with capital letters such as A, B and C. We write the elements inside curly brackets. If an element x belongs to set A, we write x ∈ A. Otherwise, we write x ∉ A.

Some standard sets have fixed symbols. N stands for natural numbers, Z for integers, Q for rational numbers and R for real numbers. Since students meet these symbols in every chapter, learn them early.

Representation of Sets

There are two ways to write a set.

Roster Form

In roster form, we list all the elements, separated by commas. For example, the set of vowels is {a, e, i, o, u}. The order does not matter, and an element is not repeated.

Set-Builder Form

In set-builder form, we state a property that every element satisfies. For example, A = {x : x is a natural number and x < 6}. We read it as “the set of all x such that x is a natural number less than 6”. As a result, the roster form of A is {1, 2, 3, 4, 5}.

Types of Sets

The chapter defines several types, and each one is a regular source of short questions.

• Empty set: it has no element, and we write it as ∅ or { }.
• Singleton set: it has exactly one element, such as {7}.
• Finite set: it has a countable number of elements, so the counting ends.
• Infinite set: it has no end to its elements, as with the set of natural numbers.
• Equal sets: they have exactly the same elements, so A = B.
• Equivalent sets: they have the same number of elements, although the elements may differ.
• Universal set: it contains all the elements under discussion, and we write it as U.

Subsets and Power Set

If every element of A is also in B, then A is a subset of B, and we write A ⊆ B. Every set is a subset of itself, and the empty set is a subset of every set. When A ⊆ B but A ≠ B, we call A a proper subset.

The power set P(A) is the set of all subsets of A. If A has n elements, then P(A) has 2ⁿ elements. For example, if A = {1, 2}, then P(A) = {∅, {1}, {2}, {1, 2}}, which has 4 elements.

Intervals as Subsets of Real Numbers

Intervals are special subsets of R. An open interval (a, b) leaves out both ends. A closed interval [a, b] includes both ends. Similarly, half-open intervals include only one end. For example, (2, 5] contains every real number greater than 2 and up to 5, including 5.

Venn Diagrams

A Venn diagram uses closed figures, usually circles, to show sets. We draw a rectangle for the universal set and a circle inside it for each set. Overlapping circles show common elements. Therefore, Venn diagrams make the operations easy to see and to check.

Operations on Sets

There are four main operations. Because each has a clear symbol, learn the symbol together with its meaning.

Union and Intersection

The union A ∪ B contains all elements that belong to A or B or both. The intersection A ∩ B contains only the elements common to both. Two sets are disjoint if A ∩ B = ∅.

For example, let A = {1, 2, 3} and B = {3, 4, 5}. Then A ∪ B = {1, 2, 3, 4, 5} and A ∩ B = {3}.

Difference and Complement

The difference A − B contains the elements that are in A but not in B. In the same example, A − B = {1, 2}. Meanwhile, the complement A′ contains all elements of U that are not in A. Therefore, A′ = U − A.

Laws of Operations

Sets follow several laws. First, the commutative laws say A ∪ B = B ∪ A and A ∩ B = B ∩ A. Next, the associative laws allow us to regroup unions or intersections. Then the distributive laws say A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C), and the other way round. Finally, De Morgan’s laws say (A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′.

Number of Elements in a Set

For a finite set A, we write n(A) for the number of elements. The key formula is n(A ∪ B) = n(A) + n(B) − n(A ∩ B). We subtract the intersection because it is counted twice. If A and B are disjoint, the last term is zero.

For three sets, the formula becomes n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(A ∩ C) + n(A ∩ B ∩ C).

Solved Examples

Example 1. Write A = {x : x is an integer and −2 < x < 3} in roster form.
Solution: the integers between −2 and 3 are −1, 0, 1 and 2. So A = {−1, 0, 1, 2}.

Example 2. How many subsets does {a, b, c, d} have?
Solution: since the set has 4 elements, it has 2⁴ = 16 subsets. Of these, 15 are proper subsets.

Example 3. In a class of 60 students, 35 like cricket and 30 like football. If 10 like both, how many like at least one game?
Solution: n(C ∪ F) = 35 + 30 − 10 = 55. Hence 55 students like at least one game, and 5 like neither.

Example 4. If U = {1, 2, 3, 4, 5, 6}, A = {1, 2, 3} and B = {3, 4}, verify (A ∪ B)′ = A′ ∩ B′.
Solution: A ∪ B = {1, 2, 3, 4}, so (A ∪ B)′ = {5, 6}. Also, A′ = {4, 5, 6} and B′ = {1, 2, 5, 6}, so A′ ∩ B′ = {5, 6}. Both sides match.

Common Mistakes in Sets Class 11

• Students write {∅} for the empty set, but {∅} has one element, namely ∅.
• Many confuse ∈ with ⊂, because ∈ links an element to a set and ⊂ links two sets.
• Some repeat elements in roster form, although a set lists each element once.
• When counting subsets, learners forget the empty set and the set itself.
• Because the complement depends on U, never find A′ without stating the universal set.

How to Revise Sets Class 11 Notes Quickly

Spend the first day on definitions, types and symbols. Next, on day two, solve the NCERT questions on subsets, power sets and intervals. Finally, on day three, draw Venn diagrams for every word problem. If you want more chapters, browse the Class 11–12 Maths notes or the wider Class 11–12 notes hub. For a similar formula-based chapter, revisit trigonometry class 10.

Wrap-Up

Sets class 11 notes show that most of the chapter depends on a handful of symbols and one counting formula. Learn the types and laws, draw a Venn diagram for each problem, and always state the universal set. Then continue with the Class 11–12 Maths category, or go back to the The Note homepage.