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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 you 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 you 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
Work through the concepts in order. Each one has an explanation, a worked example, a self-check and an exam answer.
All Concepts
All 5 concepts of the chapter, in the order the NCERT chapter teaches them.
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
The traps that cost marks in this chapter, and how its main concepts turn up in the paper.
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.
How CBSE Asks This Chapter
- Types of Relations: Reflexive, Symmetric, Transitive and Equivalence
Usually asked as: determine whether a given relation is reflexive, symmetric, transitive, or equivalence; justify with reasons. Assertion-reason questions often test whether one property implies another, which is generally false.
- Equivalence Classes and Partitions
Questions ask students to form equivalence classes for a given relation, show that the classes partition the set, or identify when two representatives give the same class.
- One-One, Onto and Bijective Functions
Often asked as a proof for a defined function, or as a graph/mapping-based classification. Long-answer questions may ask students to prove bijectivity before finding an inverse.
- Composition of Functions and Compatibility Conditions
Exam questions ask students to compute g∘f and f∘g, find a missing function from a composition, or verify associativity for given functions.
Quick Revision
Concept, formula or equation to remember, and the trap that loses marks — in one scannable view.
Types of Relations: Reflexive, Symmetric, Transitive and Equivalence
Remember
Reflexive: for every a in A, (a,a) in R.Trap to avoid
Students often call a relation reflexive after seeing one pair like (1,1).
Equivalence Classes and Partitions
Remember
Equivalence class: [a] = {x in A : x R a}.Trap to avoid
Students often write too many classes by treating every element as a new class, even when two elements are already in the same class.
One-One, Onto and Bijective Functions
Remember
One-one test: f(a) = f(b) implies a = b.Trap to avoid
Students often decide onto by looking only at the formula and forgetting the codomain.
Composition of Functions and Compatibility Conditions
Remember
Composition rule: (g∘f)(x) = g(f(x)).Trap to avoid
Students often reverse the order and compute f(g(x)) when the question asks for g(f(x)).
Invertible Functions and Finding the Inverse
Remember
Invertibility theorem: f is invertible if and only if f is bijective.Trap to avoid
Students often find an algebraic reverse formula and call it an inverse without proving or checking that the original function is bijective on the stated domain and codomain.
| Concept | Remember | Trap to avoid |
|---|---|---|
| Types of Relations: Reflexive, Symmetric, Transitive and Equivalence | Reflexive: for every a in A, (a,a) in R. | Students often call a relation reflexive after seeing one pair like (1,1). |
| Equivalence Classes and Partitions | Equivalence class: [a] = {x in A : x R a}. | Students often write too many classes by treating every element as a new class, even when two elements are already in the same class. |
| One-One, Onto and Bijective Functions | One-one test: f(a) = f(b) implies a = b. | Students often decide onto by looking only at the formula and forgetting the codomain. |
| Composition of Functions and Compatibility Conditions | Composition rule: (g∘f)(x) = g(f(x)). | Students often reverse the order and compute f(g(x)) when the question asks for g(f(x)). |
| Invertible Functions and Finding the Inverse | Invertibility theorem: f is invertible if and only if f is bijective. | Students often find an algebraic reverse formula and call it an inverse without proving or checking that the original function is bijective on the stated domain and codomain. |
Practice
Use short concept checks first, then move into the full chapter test.
Free Chapter MCQ Quiz
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