JEE Mains Maths · Relations and Functions

Sets and Counting Elements

Sets described by conditions and then combined by union, intersection and difference; counting people in overlapping groups by inclusion–exclusion; and counting subsets.

All Relations and Functions notes JEE Mains Maths strategy

Why this matters

Eighteen PYQs. Half describe two or three sets by inequalities and ask which statement about their union, intersection or difference holds. The rest count: students in overlapping groups, or subsets with a property. Three ideas cover the page.

Concept 1 of 3: Sets given by conditions, then combined

Solve each condition first, so that every set becomes an interval, a union of intervals, or a list. Then combine them on a number line: the union keeps anything in either, the intersection keeps only the overlap, and

A-B

keeps the part of

A

outside

B

. For sets of points in the plane, sketch the regions and list the points that qualify.

Definition

$A\cup B$ : in $A$ or $B$ . $A\cap B$ : in both.
$A-B=A\cap B'$ : in $A$ , not in $B$ .
$|x-a|<r$ means $a-r<x<a+r$ ; $|x-a|\ge r$ means $x\le a-r$ or $x\ge a+r$ .
A complement flips each bracket: KaTeX parse error: Unexpected character: '\' at position 2: [\̲ becomes $\ )$ and back.

Difference of sets

A-B=A\cap B'

Worked example

A=\{x:|x-1|<3\}

B=\{x:x^2\ge4\}

. Find

A-B

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2022 · 25 June 2022 · Q151Moderate

Example 1 · Relations and Functions · Sets and Counting Elements

Let

A\mathbf{= \{}x\in R:|x+ 1| < 2\}

and

B\mathbf{= \{}x\in R:|x- 1| \geq 2\}

. Then which one of the following statements is NOT true?

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.

Drill 8 more on sets given by conditions, then combined

Concept 2 of 3: Counting overlapping groups by inclusion–exclusion

Adding the sizes of groups counts everyone in two groups twice, so subtract each pairwise overlap; then everyone in all three has been added three times and subtracted three times, so add the triple overlap back. When the triple overlap is unknown, the rule that no region of the Venn diagram can be negative bounds it. People in exactly one, two or three groups can be counted from the total number of memberships.

Definition

$n(A\cup B)=n(A)+n(B)-n(A\cap B)$ .
$n(A\cup B\cup C)=\sum n(A)-\sum n(A\cap B)+n(A\cap B\cap C)$ .
Least overlap of two groups inside a total $T$ : $n(A)+n(B)-T$ .
If $s,t,u$ people are in exactly 1, 2, 3 groups: $s+t+u$ = people, $s+2t+3u$ = memberships.

Three sets

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

Worked example

Of 100 students, 60 study Hindi, 50 study Sanskrit and 30 study both. How many study neither?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2023 · 11 April 2023 · Q69Moderate

Example 2 · Relations and Functions · Sets and Counting Elements

An organization awarded 48 medals in event '

A

', 25 in event '

B

' and 18 in event '

C

'. If these medals went to total 60 men and only five men got medals in all the three events, then, how many received medals in exactly two of three events?

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.

Drill 5 more on counting overlapping groups by inclusion–exclusion

Concept 3 of 3: Counting subsets

A set of

n

elements has

2^n

subsets, because each element is either in or out. To count subsets that meet a part

B

k

elements, subtract those that avoid it:

2^n-2^{n-k}

. For an 'either … or …' condition, count the subsets that fail it and subtract.

Definition

Subsets of an $n$ -set: $2^n$ . Non-empty: $2^n-1$ . Proper and non-empty: $2^n-2$ .
Subsets meeting a $k$ -element part: $2^n-2^{n-k}$ .
Containing some fixed elements and avoiding others: $2^{\text{free elements}}$ .

Subsets meeting a k-element part

2^n-2^{n-k}

Worked example

How many subsets of

\{1,\dots,6\}

contain at least one of 1 and 2?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2022 · 26 July 2022 · Q171Moderate

Example 3 · Relations and Functions · Sets and Counting Elements

Let

\mathbf{A = \{ 1,2,3,4,5,6,7\}}

and

\mathbf{B = \{ 3,6,7,9\}}

. Then the number of elements in the set

\mathbf{\{ C \subseteq A:C \cap B \neq \phi\}}

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.

Drill 2 more on counting subsets

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

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
$2^n-2^{n-k}$

Watch out for (3)

Strict or not at the ends
→ Sets given by conditions, then combined
The least overlap is not zero
→ Counting overlapping groups by inclusion–exclusion
Only the shared elements count
→ Counting subsets

Test yourself on Relations and Functions

20 past JEE Mains questions from this chapter, timed at 48 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

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