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Relations and Functions
Relations and Functions begins by sharpening the idea of a relation from one set to another and then studies special relations on a single set. In Class 12, the most important checks are reflexive, symmetric, transitive, and equivalence relation. Equivalence relations are important because they divide a set into non-overlapping equivalence classes. This chapter expects students to connect an algebraic condition with the set of all elements related to a given element. The function part focuses on one-one, onto, bijective, composition, and inverse functions. Exam questions often test the exact condition under which a function has an inverse, not just the process of finding it. A strong answer in this chapter states the set, domain, codomain, rule, and condition clearly. Many marks are lost when students prove only one part of a property or ignore the codomain while deciding onto.
Difficulty
Medium
Study time
70-90 min
Plan by time
Pick the window that matches what you have right now.
If you have 15 min
Last-pass revision
Skim the Quick Revision table — definitions, formulas, and the traps board examiners reuse.
Open Quick RevisionIf you have 45 min
Targeted practice
Read the high-priority concepts, then take the chapter MCQ quiz to find weak spots.
Start MCQ QuizIf you have 70 min
First full pass
Walk every concept in chapter order, then revise and quiz. Best for the first time you study this chapter.
Open Key ConceptsChapter Learning Map
Start with one of the buckets below, then open the full map when you want the complete concept roadmap.
Key Concepts
Concepts grouped the way the chapter is taught — open the bucket that matches what you want to revise.
Core Concepts
high priorityOpen the chapter concepts in a clean revision order.
Types of Relations: Reflexive, Symmetric, Transitive and Equivalence
A relation R on a set A is a subset of A × A. It is reflexive if every a in A satisfies (a, a) in R, symmetric if (a, b) in R implies (b, a) in R, and transitive if (a, b) in R and (b, c) in R imply (a, c) in R. A relation that is reflexive, symmetric, and transitive is an equivalence relation.
Equivalence Classes and Partitions
If R is an equivalence relation on a set A, the equivalence class of an element a in A is [a] = {x in A : x R a}. Equivalence classes are non-empty, pairwise disjoint or identical, and together they partition A.
One-One, Onto and Bijective Functions
A function f: A to B is one-one if different elements of A have different images in B. It is onto if every element of B has a preimage in A. It is bijective if it is both one-one and onto.
Composition of Functions and Compatibility Conditions
For functions f: A to B and g: B to C, the composition g∘f is defined by (g∘f)(x) = g(f(x)) for x in A. The output of f must be suitable as an input for g.
Invertible Functions and Finding the Inverse
A function f: A to B is invertible if there exists a function f⁻¹: B to A such that f⁻¹(f(x)) = x for every x in A and f(f⁻¹(y)) = y for every y in B. A function is invertible if and only if it is bijective.
Exam Intelligence
Use this section to decide what deserves the most revision time.
High Probability Topics
- Types of Relations: Reflexive, Symmetric, Transitive and Equivalence
- Equivalence Classes and Partitions
- One-One, Onto and Bijective Functions
- Composition of Functions and Compatibility Conditions
- Invertible Functions and Finding the Inverse
Common Traps
- Checking reflexivity for only one element instead of every element in the set.
- Assuming symmetry also proves transitivity.
- Repeating equivalent classes under different representatives.
- Testing onto without using the stated codomain.
- Reversing the order in g∘f.
- Writing an inverse formula without proving or checking bijectivity.
- Ignoring restricted domain or range while deciding invertibility.
Likely Question Types
- MCQ: concept checks, applications, and common mistakes
- Very short answer: definitions, formulas, conditions, or terms
- Short answer: process, diagram, reasoning, or worked method
- Case-based: chapter scenario with linked subparts
Quick Revision
Concept, formula or equation to remember, and the trap that loses marks — in one scannable view.
- Relations are subsets of Cartesian products; special relations on a set are classified by reflexive, symmetric, and transitive tests.
- Equivalence relations create equivalence classes, and distinct equivalence classes partition the set.
- Functions are one-one when outputs are not shared and onto when the codomain is fully covered.
- A bijective function is both one-one and onto, and this is exactly the condition for invertibility.
- Composition depends on order and compatibility; g∘f means f acts first and g acts second.
- Types of Relations: Reflexive, Symmetric, Transitive and Equivalence: A relation R on a set A is a subset of A × A. It is reflexive if every a in A satisfies (a, a) in R, symmetric if (a, b) in R implies (b, a…
- Equivalence Classes and Partitions: If R is an equivalence relation on a set A, the equivalence class of an element a in A is [a] = {x in A : x R a}. Equivalence classes are n…
- One-One, Onto and Bijective Functions: A function f: A to B is one-one if different elements of A have different images in B. It is onto if every element of B has a preimage in A…
Practice
Use short concept checks first, then move into the full chapter test.
Free Chapter MCQ Quiz
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