Relations and Functions — JEE Mains Maths Formulas and Traps

Sets and Counting Elements

Sets given by conditions, then combined

Difference of sets

A-B=A\cap B'

Counting overlapping groups by inclusion–exclusion

Three sets

n(A\cup B\cup C)=\textstyle\sum n(A)-\sum n(A\cap B)+n(A\cap B\cap C)

Counting subsets

Subsets meeting a k-element part

2^n-2^{n-k}

Common traps

Strict or not at the ends

|x|<2

leaves out

\pm2

;

|x|\le2

keeps them. An option that differs from the truth only at an endpoint is usually the wrong one.

The least overlap is not zero

Two groups inside a fixed total must overlap by at least

n(A)+n(B)-T

. A bound of zero ignores the total.

Only the shared elements count

A subset of

A

meets

B

only through

A\cap B

. Elements of

B

outside

A

do not change the count.

Reflexive, Symmetric, Transitive and Equivalence Relations

Learn this subtopic in the notes

Testing reflexive, symmetric and transitive

Transitivity

aRb\ \text{and}\ bRc\ \Rightarrow\ aRc

Equivalence relations and their classes

The usual shape

aRb\iff g(a)=g(b)

Common traps

A chain back to the start

Transitivity applies to

aRb

and

bRa

too: together they force

aRa

. A symmetric relation missing

(a,a)

for such an

a

is not transitive.

Reflexive and symmetric is not enough

Many relations here are reflexive and symmetric but fail transitivity, for example

|a-b|\le1

. Test a chain that crosses the limit before calling it an equivalence.

Counting Relations and Their Elements

Learn this subtopic in the notes

Counting the pairs in a relation

Pairs in a relation

n(R)=\sum_{a}\#\{b: aRb\}

Fewest pairs to add: reflexive, symmetric, equivalence

Smallest equivalence containing R

\sum_i k_i^2\ \text{pairs, } k_i=\text{class sizes}

Counting relations of a given type

Symmetric relations

2^{n(n+1)/2}

Common traps

Count both orders

(1,2)

and

(2,1)

are two elements of a relation. A count done over unordered pairs must be doubled, except for pairs with equal entries.

Symmetry then transitivity brings the diagonal

Once

(1,2)

and

(2,1)

are both present, transitivity forces

(1,1)

and

(2,2)

. A count for 'symmetric and transitive' must include them.

Equivalences are not a power of 2

Equivalence relations correspond to partitions, not to free yes/no choices. Count the ways to split the set into classes.

Domain of a Function

Learn this subtopic in the notes

The conditions that fix a domain

Inverse-sine domain

\sin^{-1}u,\ \cos^{-1}u:\ -1\le u\le1

Nested logarithms and greatest-integer parts

Two nested logarithms

\log_a(\log_b u)\ \text{defined}\iff u>1\quad(a,b>1)

Domain of a composite function

Domain of f∘g

\{x\in\operatorname{dom}g:\ g(x)\in\operatorname{dom}f\}

Common traps

A logarithm in a denominator

\frac1{\log u}

needs

\log u\ne0

as well as

u>0

: the argument

u=1

is excluded. This is the usual missing point in a domain written as 'an interval minus a point'.

Turn integers back into intervals

[x]\le-3

means

x<-2

, not

x\le-3

: every

x

with integer part

-3

lies in

[-3,-2)

.

The inner function's gaps stay

If

g

is undefined at a point, so is

f\circ g

, whatever the simplified formula says.

Range, One-One and Onto

Learn this subtopic in the notes

Finding the range

Rational function

y=\frac{p(x)}{q(x)}\ \Rightarrow\ \Delta_x\ge0

Deciding one-one and onto

The two tests

\text{one-one: } f(a)=f(b)\Rightarrow a=b;\qquad \text{onto: } f(A)=B

Common traps

When the x² term vanishes

In the discriminant method, the value of

y

that makes the

x^2

coefficient zero turns the equation linear. Check it separately: it may or may not be in the range.

Read the codomain

The same formula can be onto one codomain and not another. A range of

(-1,1)

is not onto

(-\infty,1)

, even though every value is below 1.

Counting Functions

Learn this subtopic in the notes

Counting all functions and one-one functions

One-one functions

\frac{n!}{(n-m)!}=n(n-1)\cdots(n-m+1)

Counting onto functions

Onto functions

n^m-\binom n1(n-1)^m+\binom n2(n-2)^m-\cdots

Common traps

More inputs than outputs

A one-one function needs at least as many outputs as inputs. From a larger set to a smaller one there are none.

Add back the double-counted

Subtracting 'misses

a

' and 'misses

b

' removes the functions that miss both twice. The

+\binom n2(n-2)^m

term puts them back.

Composition, Inverse and Iterates

Learn this subtopic in the notes

Composing functions, and recovering one from the other

Composition

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

Inverse functions and self-inverse maps

Self-inverse Möbius map

f(x)=\frac{ax+b}{cx+d},\quad f\circ f=\mathrm{id}\iff a+d=0

Applying a function many times

A cycle

f^k=\mathrm{id}\ \Rightarrow\ f^{n}=f^{\,n \bmod k}

Common traps

The inner function acts first

f\circ g

means

g

first. Reading it left to right gives

g\circ f

, a different function.

Only for increasing functions

A decreasing function can meet its inverse off the line

y=x

. The shortcut

f(x)=x

is safe only when

f

is increasing.

Find the cycle on a general x

A value that repeats at one particular

x

proves nothing about the others. Show

f^k(x)=x

as an identity before reducing

n

.

Functional Equations

Learn this subtopic in the notes

Additive and multiplicative rules

Additive rule

f(x+y)=f(x)+f(y)\ \Rightarrow\ f(n)=n\,f(1)

Substitute again, then solve the pair

Two equations, two unknowns

af(x)+bf\big(g(x)\big)=h(x),\quad af\big(g(x)\big)+bf(x)=h\big(g(x)\big)

Pairing terms when f(x) + f(a − x) is constant

The standard pair

f(x)=\frac{b^x}{b^x+\sqrt b}\ \Rightarrow\ f(x)+f(1-x)=1

Common traps

A constant term changes f(0)

In

f(x+y)=f(x)+f(y)+c

, putting

x=y=0

gives

f(0)=-c

, not 0. Find it before using any pattern.

A sum may need no solving

If the question asks for

f(a)+f(b)

where

b

is

a

's partner, putting

x=a

and

x=b

and adding the two equations can give it directly, without finding

f

.

Count the terms

An odd number of terms leaves a middle term unpaired. Add it separately; forgetting it is the usual gap between two options.

Relations and Functions formulas

Sets and Counting Elements

Sets given by conditions, then combined

Counting overlapping groups by inclusion–exclusion

Counting subsets

Common traps

Reflexive, Symmetric, Transitive and Equivalence Relations

Testing reflexive, symmetric and transitive

Equivalence relations and their classes

Common traps

Counting Relations and Their Elements

Counting the pairs in a relation

Fewest pairs to add: reflexive, symmetric, equivalence

Counting relations of a given type

Common traps

Domain of a Function

The conditions that fix a domain

Nested logarithms and greatest-integer parts

Domain of a composite function

Common traps

Range, One-One and Onto

Finding the range

Deciding one-one and onto

Common traps

Counting Functions

Counting all functions and one-one functions

Counting onto functions

Common traps

Composition, Inverse and Iterates

Composing functions, and recovering one from the other

Inverse functions and self-inverse maps

Applying a function many times

Common traps

Functional Equations

Additive and multiplicative rules

Substitute again, then solve the pair

Pairing terms when f(x) + f(a − x) is constant

Common traps

More JEE Mains Maths formula sheets