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Schooling · Class 11 · Mathematics · Relations and Functions · Chapter on a page

CBSE Class 11 · Mathematics · NCERT Mathematics

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.

Fact-checked on 21 Oct 2026: an AI-assisted check of this note against cited sources; the errors it found were corrected that day. Some errors may remain.

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.

Mind map of Relations and Functions: Key ideas, Formulas and facts, Watch out forKey ideasAn ordered pair (a, b) is a pair of…The Cartesian product A × B of two…A relation R from set A to set B is…A function f from set A to set B…The domain of a function is the set…Special types of functions include…Algebra of functionsFormulas and factsA × B = {(aFor any sets, A × B ≠ B × A in…A × (B ∪ C) = (A × B) ∪ (A × C) and…The number of relations from A to B…A function requiresDomain of f(x) = 1/(x − 2) is all…Domain of f(x) = √(x − 3) is all…Watch out forConfusing ordered pairs with setsThinking every relation is a functionConfusing codomain with rangeForgetting to exclude values that…Writing A × B = B × ARelations andFunctions
The mind map as a list
  • Key ideas
    • An ordered pair (a, b) is a pair of…
    • The Cartesian product A × B of two…
    • A relation R from set A to set B is…
    • A function f from set A to set B…
    • The domain of a function is the set…
    • Special types of functions include…
    • Algebra of functions
  • Formulas and facts
    • A × B = {(a
    • For any sets, A × B ≠ B × A in…
    • A × (B ∪ C) = (A × B) ∪ (A × C) and…
    • The number of relations from A to B…
    • A function requires
    • Domain of f(x) = 1/(x − 2) is all…
    • Domain of f(x) = √(x − 3) is all…
  • Watch out for
    • Confusing ordered pairs with sets
    • Thinking every relation is a function
    • Confusing codomain with range
    • Forgetting to exclude values that…
    • Writing A × B = B × A

What this chapter is about

This chapter builds directly on your understanding of sets from the previous chapter. Here you learn how to connect elements of one set to elements of another set in a structured way. A relation pairs elements according to some rule, and a function is a special kind of relation where each input gives exactly one output. These ideas form the backbone of all higher mathematics.

Key ideas

  • An ordered pair (a, b) is a pair of elements where order matters: (a, b) is different from (b, a) unless a = b.
  • The Cartesian product A × B of two sets A and B is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B. If A has m elements and B has n elements, then A × B has m × n elements.
  • A relation R from set A to set B is any subset of A × B. The set of all first elements in R is called the domain, B is called the codomain, and the set of all second elements that actually appear in R is called the range.
  • A function f from set A to set B (written f: A → B) is a relation where every element of A is related to exactly one element of B. No element of A is left out, and no element of A is paired with two different elements of B.
  • The domain of a function is the set of all permissible inputs, and the range is the set of all outputs that actually occur.
  • Special types of functions include the identity function f(x) = x, constant functions f(x) = c, polynomial functions, rational functions, and the modulus function f(x) = |x|.
  • Algebra of functions: if f and g are real functions with a common domain, then (f + g)(x) = f(x) + g(x), (f − g)(x) = f(x) − g(x), (fg)(x) = f(x) × g(x), and (f/g)(x) = f(x)/g(x) provided g(x) ≠ 0.

Formulas and facts to remember

  • Cartesian product: A × B = {(a, b) : a ∈ A and b ∈ B}. The number of elements is n(A) × n(B).
  • For any sets, A × B ≠ B × A in general (order matters in ordered pairs).
  • A × (B ∪ C) = (A × B) ∪ (A × C) and A × (B ∩ C) = (A × B) ∩ (A × C).
  • The number of relations from A to B is 2^(m×n) if A has m elements and B has n elements.
  • A function requires: every element of the domain has an image, and each element of the domain has only one image.
  • Domain of f(x) = 1/(x − 2) is all real numbers except x = 2.
  • Domain of f(x) = √(x − 3) is all real numbers x ≥ 3.
  • Range of f(x) = x² for x ∈ R is [0, ∞).

Common mistakes

  • Confusing ordered pairs with sets: (3, 5) is not the same as (5, 3), but {3, 5} equals {5, 3} → Remember that order matters in ordered pairs but not in sets.
  • Thinking every relation is a function → Check that each domain element maps to exactly one codomain element.
  • Confusing codomain with range → Codomain is the set where outputs are allowed to land; range is the set of outputs that actually occur.
  • Forgetting to exclude values that make denominators zero → Always check where the denominator equals zero and remove those from the domain.
  • Writing A × B = B × A → The Cartesian product is not commutative; the pairs differ in order.

Quick revision

  • Ordered pair: (a, b) ≠ (b, a) unless a = b.
  • A × B has n(A) × n(B) ordered pairs.
  • A function assigns exactly one output to each input in the domain.
  • Domain is where the function is defined; range is the set of actual outputs.
  • For √(expression), the expression inside must be ≥ 0; for 1/(expression), the expression must ≠ 0.
  • Algebra of functions: add, subtract, multiply or divide the output values, with common domain.