Schooling · Class 12 · Mathematics · Relations and Functions · Chapter on a page
CBSE Class 12 · Mathematics · NCERT Mathematics Part-I
Relations and Functions: 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
- relation
- reflexive
- symmetric
- transitive
- equivalence relation
- A function f
- onto (surjective)
- Formulas and facts
- (g ∘ f)(x) = g(f(x))
- f⁻¹(f(x)) = x for all x in domain of f
- f(f⁻¹(y)) = y for all y in range of f
- Formula or Rule
- (g ∘ f)⁻¹ = f⁻¹ ∘ g⁻¹
- Worked examples
- Checking whether a relation is an…
- Determining if a function is one-one…
- Finding the inverse of a bijective…
- Watch out for
- Forgetting to check reflexivity at…
- Confusing one-one with onto
- Assuming every function has an inverse
- Writing composition in wrong order
- Believing a relation that is…
What this chapter is about
In Class 11, you met the basic ideas of relations and functions. This chapter takes those ideas further by asking deeper questions: when is a relation an equivalence relation, and when is a function invertible? These are foundational concepts that appear throughout higher mathematics, from abstract algebra to real analysis.
Key ideas
- A relation R on a set A is a subset of A × A; we write a R b to mean (a, b) belongs to R.
- A relation is reflexive if every element is related to itself: (a, a) ∈ R for all a in A.
- A relation is symmetric if whenever a R b, we also have b R a.
- A relation is transitive if whenever a R b and b R c, we also have a R c.
- An equivalence relation is one that is reflexive, symmetric and transitive; it partitions the set into disjoint equivalence classes.
- A function f : A → B is one-one (injective) if distinct elements in A map to distinct elements in B: f(a₁) = f(a₂) implies a₁ = a₂.
- A function is onto (surjective) if every element in B is the image of at least one element in A.
- A function is bijective if it is both one-one and onto; only bijective functions have inverses.
Formulas and facts to remember
- Formula or Rule: (g ∘ f)(x) = g(f(x)) · Meaning: Composition: first apply f, then apply g to the result
- Formula or Rule: f⁻¹(f(x)) = x for all x in domain of f · Meaning: The inverse undoes the function
- Formula or Rule: f(f⁻¹(y)) = y for all y in range of f · Meaning: Applying f to its inverse returns the original value
- Formula or Rule: If f : A → B and g : B → C are both one-one, then g ∘ f is one-one · Meaning: One-one property is preserved under composition
- Formula or Rule: If f : A → B and g : B → C are both onto, then g ∘ f is onto · Meaning: Onto property is preserved under composition
- Formula or Rule: (g ∘ f)⁻¹ = f⁻¹ ∘ g⁻¹ · Meaning: Inverse of a composition reverses the order
- Formula or Rule: Number of reflexive relations on a set with n elements = 2^(n² − n) · Meaning: Diagonal pairs must be present; other pairs are optional
Common mistakes
- Forgetting to check reflexivity at every element → always verify (a, a) ∈ R for each a in the set, not just one.
- Confusing one-one with onto → one-one is about no two inputs sharing an output; onto is about every output being hit.
- Assuming every function has an inverse → only bijective functions are invertible; check both conditions first.
- Writing composition in wrong order: thinking (g ∘ f)(x) means f(g(x)) → remember g ∘ f means apply f first, then g.
- Believing a relation that is symmetric and transitive must be reflexive → this is false; you need an independent check.
Quick revision
- Equivalence relation = reflexive + symmetric + transitive; it partitions a set into disjoint classes.
- One-one means different inputs give different outputs; onto means every output is achieved.
- Only bijective functions have inverses; to find f⁻¹, solve y = f(x) for x.
- Composition order: (g ∘ f)(x) = g(f(x)) — f acts first.
- Inverse of a composition: (g ∘ f)⁻¹ = f⁻¹ ∘ g⁻¹.