Relations and Functions

Chapter overview

Relations and functions describe how elements of one set are associated with elements of another set. In Class 12, the focus is mainly on types of relations, equivalence relations, one-one functions, onto functions, composition, and invertible functions.

1. Cartesian product

If $A$ and $B$ are two non-empty sets, then:

$$A\times B=\{(a,b):a\in A,\ b\in B\}$$

If:

$$n(A)=m,\qquad n(B)=n$$

then:

$$n(A\times B)=mn$$

Usually:

$$A\times B\ne B\times A$$

2. Relation

A relation $R$ from set $A$ to set $B$ is a subset of $A\times B$.

$$R\subseteq A\times B$$

If $(a,b)\in R$, then $a$ is related to $b$.

Notation:

$$aRb$$

Number of possible relations from $A$ to $B$:

$$2^{mn}$$

where:

$$n(A)=m,\qquad n(B)=n$$

3. Domain, codomain, and range

For a relation $R$ from $A$ to $B$:

Domain:

$$\text{Domain}(R)=\{a\in A:(a,b)\in R\text{ for some }b\in B\}$$

Range:

$$\text{Range}(R)=\{b\in B:(a,b)\in R\text{ for some }a\in A\}$$

Codomain:

$$B$$

Important:

$$\text{Range}\subseteq\text{Codomain}$$

4. Types of relations on a set

Let $R$ be a relation on a set $A$.

Reflexive relation

$$(a,a)\in R \quad \forall a\in A$$

Symmetric relation

$$(a,b)\in R \Rightarrow (b,a)\in R$$

Transitive relation

$$(a,b)\in R \text{ and } (b,c)\in R \Rightarrow (a,c)\in R$$

Equivalence relation

A relation is an equivalence relation if it is:

• reflexive,
• symmetric,
• transitive.

5. Function

A function $f$ from $A$ to $B$ is a relation in which every element of $A$ has exactly one image in $B$.

$$f:A\rightarrow B$$

If $x\in A$, then image of $x$ is:

$$f(x)$$

Function condition:

$$x_1=x_2 \Rightarrow f(x_1)=f(x_2)$$

6. One-one function

A function is one-one or injective if different elements of the domain have different images.

$$f(x_1)=f(x_2) \Rightarrow x_1=x_2$$

Equivalent:

$$x_1\ne x_2 \Rightarrow f(x_1)\ne f(x_2)$$

7. Onto function

A function is onto or surjective if every element of codomain has a preimage in domain.

$$\text{Range}(f)=\text{Codomain}$$

Meaning:

$$\forall y\in B,\ \exists x\in A \text{ such that } f(x)=y$$

8. Bijective function

A function is bijective if it is both one-one and onto.

9. Composition of functions

If:

$$f:A\rightarrow B$$

and:

$$g:B\rightarrow C$$

then:

$$g\circ f:A\rightarrow C$$

defined by:

$$(g\circ f)(x)=g(f(x))$$

In general:

$$g\circ f\ne f\circ g$$

Associative property:

$$h\circ(g\circ f)=(h\circ g)\circ f$$

10. Invertible functions

A function $f:A\rightarrow B$ is invertible if there exists a function $g:B\rightarrow A$ such that:

$$g\circ f=I_A$$

and:

$$f\circ g=I_B$$

Then:

$$g=f^{-1}$$

A function is invertible if and only if it is bijective.

$$f\text{ is invertible} \iff f\text{ is one-one and onto}$$

11. Solved examples

Example 1

Let:

$$A=\{1,2,3\}$$

and:

$$R=\{(1,1),(2,2),(3,3)\}$$

Then $R$ is reflexive because every element is related to itself.

Example 2

Check whether:

$$f:R\rightarrow R,\quad f(x)=2x+3$$

is one-one.

If:

$$f(x_1)=f(x_2)$$

then:

$$2x_1+3=2x_2+3$$ $$x_1=x_2$$

So $f$ is one-one.

Example 3

Find inverse of:

$$f(x)=2x+3$$

Let:

$$y=2x+3$$

Then:

$$x=\frac{y-3}{2}$$

Therefore:

$$f^{-1}(x)=\frac{x-3}{2}$$

12. Common mistakes

• Confusing relation and function.
• Thinking every relation is a function.
• Confusing range and codomain.
• Forgetting that onto means range equals codomain.
• Assuming $f\circ g=g\circ f$.
• Saying a function is invertible without checking one-one and onto.

13. Quick revision