Knowledge organisers

Relations and Functions: CBSE Class 12 Maths knowledge organiser

Everything to know about relations and functions on one page: key definitions, the formulas, a worked example, the mistakes to avoid and a checklist of what you should be able to do.

Download the A4 sheet (PDF)

Key definitions

Reflexive, symmetric, transitive: The three properties that together make an equivalence relation.
One-one function: Different inputs always give different outputs.
Onto function: Every element of the codomain is the image of some element of the domain.

Key formulas

Reflexive \((a,a)\in R\ \forall a\); symmetric \((a,b)\in R\Rightarrow(b,a)\in R\); transitive \((a,b),(b,c)\in R\Rightarrow(a,c)\in R\); equivalence = all three
One-one: \(f(x_1)=f(x_2)\Rightarrow x_1=x_2\). Onto: range = codomain. Bijective: both
Counting, \(\|A\|=m\), \(\|B\|=n\)	\(\text{relations on }A:\ 2^{m^2};\) \(\text{functions }A\to B:\ n^m\)
One-one \(A\to B\) (\(m\le n\)); bijections (\(m=n\))	\(\frac{n!}{(n-m)!};\) \(n!\)

Worked example

On \(\{1, \ldots, 6\}\), \(a\,R\,b \iff 3 \mid (a - b)\): classes \(\{1, 4\}, \{2, 5\}, \{3, 6\}\).

Common mistakes

Checking reflexivity for only the elements that appear in \(R\) instead of all of \(A\).
Showing onto by solving for \(x\) but not checking that \(x\) lies in the domain (e.g. \(x \in \mathbb{N}\), \(x \ne 2\)).
Forgetting that the answer depends on the domain and codomain: \(x^2\) is not one-one on \(\mathbb{R}\) but is one-one on \([0, \infty)\).
Giving an “example” instead of a proof when asked to show a property holds; a single counter-example is enough only to show it fails.

You should be able to…

Test a relation for reflexive, symmetric and transitive and tell whether a function is one-one or onto.
Prove equivalence relations, write equivalence classes, count relations and functions, and prove one-one or onto properly.
Write the 5-mark relation and function proofs in full, with a heading and reason for each property and counter-examples where needed.
Handle bijection proofs on restricted domains, tricky counting of relations and the hardest assertion-reason items.

The printable sheet

Relations and Functions knowledge organiser for CBSE Class 12 Maths: one A4 page of key definitions, formulas, a worked example and common mistakes — Relations and Functions knowledge organiser (CBSE Class 12 Maths), A4. Download the PDF.

Revise it next

Other CBSE Class 12 Maths topics: Inverse Trigonometric Functions · Matrices · Determinants · Continuity and Differentiability · Application of Derivatives · Integrals · Application of Integrals · Differential Equations · Vector Algebra · Three Dimensional Geometry · Linear Programming · Probability · All CBSE Class 12 Maths organisers