Schooling · Class 11 · Mathematics · Sets · Chapter on a page
CBSE Class 11 · Mathematics · NCERT Mathematics
Sets: chapter on a page
A cut of Shishya's notes for this chapter: the first paragraph of the introduction and the key ideas, formulas and facts, common mistakes and quick revision lists, word for word. Nothing is added or reworded; the worked examples are left out (they are on the chapter page).
About the notes: Written by Shishya's AI on 27 Sept 2026 from the chapter's title and class level, in Shishya's own words — not a copy or summary of the textbook. Read the official chapter for the book's own text, activities and exercises.
This note's current text has no applied fact-check on record.
Mind map
Drawn from the same notes by a fixed rule (no AI): each label is the item's own words — its bold term, its label before a colon, a short formula as written, or its first words.
The mind map as a list
- Key ideas
- A set is well-defined when, for any…
- Two ways to describe a set
- A set with no elements is the empty…
- If every element of set A is also in…
- Two sets are equal if and only if…
- The power set of A, written P(A), is…
- Union (A ∪ B) contains elements in A…
- Formulas and facts
- n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
- Formula / Rule
- n(P(A)) = 2ⁿ where n = n(A)
- (A′)′ = A
- A ∪ A′ = U and A ∩ A′ = ∅
- A − B = A ∩ B′
- Watch out for
- Writing the empty set as {∅} instead…
- Confusing ∈ (element of) with ⊆…
- Forgetting to subtract the…
- Assuming A − B = B − A
- Listing duplicate elements in roster…
What this chapter is about
Sets form the foundation of modern mathematics. A set is a well-defined collection of distinct objects, called elements or members. In Class 11, you study sets because nearly every branch of mathematics—relations, functions, probability, sequences—uses the language of sets. Understanding sets now prepares you for all senior secondary mathematics.
Key ideas
- A set is well-defined when, for any object, we can decide with certainty whether it belongs to the set or not. For example, the collection of all vowels in the English alphabet is well-defined; the collection of tall students in a class is not.
- Two ways to describe a set: roster (or tabular) form lists every element inside braces, such as {a, e, i, o, u}; set-builder form states a property, such as {x : x is a vowel in the English alphabet}.
- A set with no elements is the empty set (or null set), written ∅ or {}. A set with a finite number of elements is a finite set; otherwise it is infinite.
- If every element of set A is also in set B, then A is a subset of B, written A ⊆ B. Every set is a subset of itself, and ∅ is a subset of every set.
- Two sets are equal if and only if each is a subset of the other, that is, A = B when A ⊆ B and B ⊆ A.
- The power set of A, written P(A), is the set of all subsets of A. If A has n elements, P(A) has 2ⁿ elements.
- Union (A ∪ B) contains elements in A or B or both. Intersection (A ∩ B) contains elements in both A and B. Difference (A − B) contains elements in A but not in B. Complement (A′) contains elements in the universal set U but not in A.
- Venn diagrams represent sets as circles inside a rectangle (the universal set) and help visualise operations and relationships.
Formulas and facts to remember
- Formula / Rule: n(A ∪ B) = n(A) + n(B) − n(A ∩ B) · Meaning: Count of union equals sum of counts minus overlap.
- Formula / Rule: n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(A ∩ C) + n(A ∩ B ∩ C) · Meaning: Inclusion-exclusion for three sets.
- Formula / Rule: n(P(A)) = 2ⁿ where n = n(A) · Meaning: Number of subsets of a set with n elements.
- Formula / Rule: (A′)′ = A · Meaning: Complement of complement returns the original set.
- Formula / Rule: A ∪ A′ = U and A ∩ A′ = ∅ · Meaning: A set and its complement together make U; they share nothing.
- Formula / Rule: De Morgan's laws: (A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′ · Meaning: Complement of union is intersection of complements, and vice versa.
- Formula / Rule: A − B = A ∩ B′ · Meaning: Difference can be written as intersection with complement.
Common mistakes
- Writing the empty set as {∅} instead of ∅ or {} → {∅} is a set containing one element (the empty set), not the empty set itself.
- Confusing ∈ (element of) with ⊆ (subset of) → 3 ∈ {1, 2, 3} but {3} ⊆ {1, 2, 3}; an element is not the same as a singleton subset.
- Forgetting to subtract the intersection when counting a union → always use n(A ∪ B) = n(A) + n(B) − n(A ∩ B).
- Assuming A − B = B − A → set difference is not commutative; the two results usually differ.
- Listing duplicate elements in roster form → each element appears only once in a set; {2, 2, 3} should be {2, 3}.
Quick revision
- A set is well-defined; order and repetition do not matter.
- ∅ is a subset of every set; every set is a subset of itself.
- Power set of n elements has 2ⁿ subsets.
- Union adds, intersection keeps common, difference removes.
- De Morgan: complement of union = intersection of complements, and vice versa.
- Inclusion-exclusion prevents double-counting in union problems.