Mathematics · Sets and Functions · Chapter notes
Sets · Class 11 Maths Notes
Class 11 notes on Sets: notation, types of sets, the power set, operations with Venn diagrams, De Morgan's laws and the cardinality formula, with worked examples and a formula sheet.
In short
A set is a well-defined collection, and that phrase is doing real work: membership must be decidable, so the collection of tall people is not a set. Everything else in the chapter is notation for combining sets, plus one counting formula, n(A∪B) = n(A) + n(B) − n(A∩B), that removes the double count.
Contents
What Is a Set?
A set is simply a well-defined collection of distinct objects. That word well-defined is the whole idea: given any object, there must be an unambiguous yes or no answer to whether it belongs. Sets are the language in which all of modern mathematics is written.
Sets are named with capital letters and their members listed in braces. If a is a member of A we write a ∈ A; if not, a ∉ A. Order does not matter and repetition is ignored: {1, 2, 2, 3} is the same set as {1, 2, 3}.
• Roster (tabular) form: list every element, for example A = {2, 4, 6, 8}.
• Set-builder form: state the defining property, for example A = {x : x is an even natural number less than 10}. Set-builder is essential when a set is infinite or too large to list.
Types of Sets
A handful of named sets appear again and again. Knowing them removes most of the confusion in this chapter.
| Set | Meaning | Example |
|---|---|---|
| Empty (null) set ∅ | Has no elements | {x : x² = −1, x real} |
| Singleton | Exactly one element | {5} |
| Finite set | Countable, ends | {1, 2, 3} |
| Infinite set | Never ends | N, Z, R |
| Equal sets | Same elements | {1,2}={2,1} |
| Equivalent sets | Same number of elements | {a,b} & {1,2} |
A is a subset of B (written A ⊆ B) if every element of A is also in B. If A ⊆ B but A ≠ B, then A is a proper subset (A ⊂ B). The empty set is a subset of every set, and every set is a subset of itself.
The Power Set and Counting Subsets
Collecting all subsets of a set produces a new, larger set, and counting them gives a clean formula that JEE loves.
The power set P(A) is the set of all subsets of A, including ∅ and A itself. If A has n elements, then:
number of proper subsets = 2n − 1
number of non-empty subsets = 2n − 1
Each element is either in or out of a given subset, two choices each, giving 2n combinations.
If A = {a, b, c}, then n = 3, so A has 2³ = 8 subsets and 8 − 1 = 7 proper subsets. For a set of 5 elements there are 2⁵ = 32 subsets and 31 proper subsets.
Operations on Sets
Sets combine through three basic operations. A Venn diagram makes each one obvious.
• Union A ∪ B: everything in A or B (or both).
• Intersection A ∩ B: only what is in both.
• Difference A − B: in A but not in B.
• Complement A': everything in the universal set U that is not in A.
Two identities that convert complements of combinations, used constantly in logic and probability:
(A ∩ B)' = A' ∪ B'
In words: the complement of a union is the intersection of the complements, and vice versa. The operation flips when you take the complement.
The Cardinality Formula
Counting the elements of a union is one of the most common exam applications, especially in word problems.
n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A∩B) − n(B∩C) − n(C∩A) + n(A∩B∩C)
When you add n(A) and n(B), the elements in both sets get counted twice, so you subtract n(A ∩ B) once to correct it. This is the inclusion-exclusion principle, and the three-set version simply extends the same idea.
In a class, 30 students like tea, 25 like coffee, and 12 like both. How many like at least one? n(T ∪ C) = 30 + 25 − 12 = 43. If the class has 50 students, then 50 − 43 = 7 like neither.
Sets: One-Page Formula Sheet
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