Class 12 Mathematics Chapter 1 – Revision Notes: Relations and Functions
- Relation: a subset R of A×A; write aRb to mean (a,b)∈R.
- Reflexive: (a,a)∈R for every a∈A.
- Symmetric: (a,b)∈R ⇒ (b,a)∈R.
- Transitive: (a,b)∈R and (b,c)∈R ⇒ (a,c)∈R.
- Equivalence relation: reflexive + symmetric + transitive together. Partitions the set into disjoint equivalence classes.
- One-one (injective): f(x1)=f(x2) ⇒ x1=x2.
- Onto (surjective): range(f) = co-domain.
- Bijective: both one-one and onto — the only kind of function that is invertible.
- Key results: if f:A→B and g:B→C are both bijective, gof:A→C is bijective and (gof)⁻¹=f⁻¹og⁻¹.
- Standard non-injective/non-surjective functions on R→R: Modulus |x|, Greatest Integer [x], Signum function.
- Binary operation: a function *:A×A→A. Commutative if a*b=b*a; associative if (a*b)*c=a*(b*c); identity e satisfies a*e=e*a=a; a is invertible if some b gives a*b=b*a=e.
- Syllabus note (2026-27): 2 exercises + Miscellaneous — 1.1 (16Q), 1.2 (12Q), Miscellaneous (19Q), 47 questions total.
Related
- Chapter 2: Inverse Trigonometric Functions – Revision Notes
- Chapter 3: Matrices – Revision Notes
- Chapter 4: Determinants – Revision Notes
- Chapter 5: Continuity and Differentiability – Revision Notes
- Chapter 6: Applications of Derivatives – Revision Notes
- Chapter 7: Integrals – Revision Notes
- Chapter 8: Applications of Integrals – Revision Notes
- Chapter 9: Differential Equations – Revision Notes
- Chapter 10: Vectors – Revision Notes
- Chapter 11: Three-Dimensional Geometry – Revision Notes
- Chapter 12: Linear Programming – Revision Notes
- Chapter 13: Probability – Revision Notes
Frequently Asked Questions
What is the difference between a relation and a function?
A relation is simply any set of ordered pairs connecting elements of two sets, while a function is a special type of relation where every element of the domain is associated with exactly one element of the range.
Last Updated: September 23, 2026
What are one-one and onto functions?
A one-one (injective) function maps distinct elements of the domain to distinct elements of the range, an onto (surjective) function covers every element of the range, and a bijective function is both, which is essential for a function to have an inverse.
Chapter Quiz — Test Your Understanding
Class 12 Mathematics Chapter 1 – Solutions and Important Questions
Need full answers or more practice? See the Class 12 Mathematics Chapter 1 Solutions and Class 12 Mathematics Chapter 1 Extra Questions. These Class 12 Mathematics Chapter 1 notes are ideal for quick revision just before exams.
Common Mistakes Students Make in Relations and Functions
- Skipping a property when checking equivalence relations: verifying reflexive and symmetric but forgetting to check transitive (or vice versa).
- Mixing up one-one and onto checks: not testing each with its correct method — one-one needs different inputs to give different outputs, onto needs every codomain element to have a preimage.
- Domain errors: not restricting the domain correctly for functions not defined everywhere (division, square roots).
- Reversing function composition: computing f(g(x)) as g(f(x)).
What to Revise First (and Last) in This Chapter
Prioritise the equivalence-relation checklist (reflexive, symmetric, transitive) and the one-one/onto definitions first, since they anchor most proof-based questions. Leave detailed composition-of-functions and binary-operation questions for an earlier, slower revision pass.
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