Chapter 1 · Sets

1.1 Sets and Their Representations

A set is a well-defined collection of distinct objects. Describe it by listing its elements (roster form) or by a property they share (set-builder form): $\{1,2,3,4\}=\{x : x\in\mathbb{N},\ x\lt 5\}$.

The squad of an IPL team, the vowels of the English alphabet, the prime numbers, the solutions of $x^2-5x+6=0$, the points of a circle — mathematics is full of collections. Very often what matters is not any single object but the whole collection: which numbers solve an equation, which inputs a function accepts, which outcomes of a dice throw count as "even". To talk about collections precisely we need a small, exact language, and that language is the theory of sets.

Sets were introduced in the 1870s by the German mathematician Georg Cantor, and within a few decades they had become the common language of every branch of mathematics. You will use them all through this course: the domain and range of a function (Chapter 2) are sets, the solution of an inequality (Chapter 5) is a set written as an interval, and in probability (Chapter 14) an event is simply a set of outcomes. In this section we learn what a set is, the two standard ways of writing one down, and how to recognise empty, finite, infinite and equal sets.

What Is a Set?

In everyday speech "a collection of good films" is a perfectly sensible phrase. In mathematics it is not acceptable, because two people can disagree about whether a particular film belongs to it. A mathematical collection must come with a rule that settles membership for every object, with no room for opinion.

📘 Definition 1.1 — Set, element

A set is a well-defined collection of objects. Well-defined means that for any object whatever, we can decide definitely whether or not it belongs to the collection. The objects in a set are called its elements or members.

Sets are usually named by capital letters $A, B, X,\dots$ and elements by small letters. If $a$ is an element of $A$ we write $a\in A$ ("$a$ belongs to $A$"); if not, we write $a\notin A$ ("$a$ does not belong to $A$").

A few sets of numbers are used so often that they have permanent names. Learn these symbols now — they appear on almost every page of this course.

Symbol	Set	Elements
$\mathbb{N}$	natural numbers	$1,2,3,4,\dots$ (in this course $0\notin\mathbb{N}$)
$\mathbb{Z}$	integers	$\dots,-3,-2,-1,0,1,2,3,\dots$
$\mathbb{Q}$	rational numbers	numbers $\dfrac pq$ with $p,q\in\mathbb{Z}$, $q\ne0$, e.g. $-\tfrac34,\ 0.25,\ 7$
$\mathbb{R}$	real numbers	all points of the number line: rationals and irrationals such as $\sqrt2$, $\pi$
$\mathbb{Z}^+,\ \mathbb{Q}^+,\ \mathbb{R}^+$	positive integers, rationals, reals	the members of $\mathbb{Z},\ \mathbb{Q},\ \mathbb{R}$ that are greater than $0$

✏️ Example 1 — Is it a set? Does it belong?

(a) Which of these collections are sets? (i) the odd natural numbers less than $10$; (ii) the tall students in your class; (iii) the students in your class whose height is more than $160$ cm; (iv) the most interesting chapters of this course.
(b) Insert $\in$ or $\notin$: $\ 7\ \square\ \mathbb{N}$, $\ -2\ \square\ \mathbb{N}$, $\ \tfrac12\ \square\ \mathbb{Z}$, $\ \sqrt2\ \square\ \mathbb{Q}$, $\ 0\ \square\ \mathbb{Z}$.

💡 Show solution▶

(a) Ask of each one: could two careful people disagree about membership? (i) No — the members are exactly $1,3,5,7,9$. It is a set. (ii) Yes — "tall" has no agreed meaning, so this is not a set. (iii) No — a tape measure settles every case. It is a set. (iv) Yes — "interesting" is a matter of opinion, so this is not a set.

(b) $7\in\mathbb{N}$; $\ -2\notin\mathbb{N}$ (natural numbers are positive); $\ \tfrac12\notin\mathbb{Z}$ (not a whole number); $\ \sqrt2\notin\mathbb{Q}$ ($\sqrt2$ is irrational — you proved this in Class 10); $\ 0\in\mathbb{Z}$.

Lesson Compare (ii) and (iii): turning a vague word into a measurable condition is exactly how a mathematician makes a collection well-defined.

Roster Form and Set-Builder Form

There are two standard ways to write a set down. We can simply list its elements, or we can state a property that its elements, and only its elements, possess.

📘 Definition 1.2 — Roster form and set-builder form

Roster (tabular) form: all the elements are listed, separated by commas, inside braces $\{\ \}$. For example, the set of even natural numbers less than $10$ is $\{2,4,6,8\}$.

Set-builder form: the set is described by a property $P(x)$ shared by all its elements and by no other object: $$\{x : P(x)\},$$ read "the set of all $x$ such that $P(x)$". For example, $\{2,4,6,8\}=\{x : x\in\mathbb{N},\ x \text{ is even},\ x\lt10\}$. A vertical bar is sometimes used instead of the colon: $\{x\mid P(x)\}$.

⚠️ Three rules of the roster form

1. Order does not matter. $\{1,2,3\}$, $\{3,1,2\}$ and $\{2,3,1\}$ are the same set.

2. Repetition does not count. An element is either in a set or not; listing it twice adds nothing. The letters of the word "SCHOOL" form the set $\{S,C,H,O,L\}$ — write each element once.

3. Dots need a clear pattern. $\{1,3,5,7,\dots\}$ is acceptable because the pattern (odd natural numbers) is unmistakable; $\{1,3,\dots\}$ is not — it could equally mean $1,3,9,27,\dots$ An ellipsis is also used for long finite lists, as in $\{1,2,3,\dots,100\}$.

✏️ Example 2 — From one form to the other

Write (a) $\{x : x\in\mathbb{Z},\ -3\lt x\le2\}$ and (b) $\{x : x\in\mathbb{N},\ x \text{ is a divisor of } 18\}$ in roster form; (c) $\{1,4,9,16,25\}$ in set-builder form; (d) the set of letters of the word "COMMITTEE" in roster form.

💡 Show solution▶

(a) The integers greater than $-3$ and at most $2$. Note that $-3$ itself is excluded ($\lt$) and $2$ is included ($\le$): $\{-2,-1,0,1,2\}$.

(b) Test $1,2,\dots,18$ — or pair the divisors: $18=1\times18=2\times9=3\times6$. So the set is $\{1,2,3,6,9,18\}$.

(c) The elements are the squares $1^2,2^2,3^2,4^2,5^2$: $\{x : x=n^2,\ n\in\mathbb{N},\ n\le5\}$. (Also correct: $\{x : x \text{ is the square of a natural number},\ x\le25\}$.)

(d) The letters C, O, M, M, I, T, T, E, E with repeats written once: $\{C,O,M,I,T,E\}$ — six elements, although the word has nine letters.

Check In (a), substitute the end points: $-3\lt-3$ is false (so $-3$ is out) and $2\le2$ is true (so $2$ is in). ✓

🧭 Problem-solving strategy — converting between the forms

Set-builder → roster. Identify the "home" set the variable runs over ($\mathbb{N}$, $\mathbb{Z}$, $\mathbb{R}$, letters…). Test the candidates one by one, or solve the condition (an equation, an inequality) and then keep only the solutions that lie in the home set.
Pay attention to strict ($\lt$, $\gt$) versus non-strict ($\le$, $\ge$) inequalities at the end points.
Roster → set-builder. Look for the pattern: multiples, squares, divisors, a formula in $n$. Write the formula, say where $n$ comes from, and give its range.
Check by regenerating the roster from your set-builder description.

A set-builder description is a test. The machine takes the candidates from the home set one at a time, applies the property, and keeps those that pass (blue) while rejecting the rest (grey). The roster form builds up in the readout. Watch what happens when nothing passes, and when the pattern carries on beyond the window.

Set:

testing $x$—

passes the test?—

roster form so far—

$n(A)$ so far—

The Empty Set, Finite and Infinite Sets

Consider $\{x : x\in\mathbb{N},\ 1\lt x\lt2\}$. There is no natural number strictly between $1$ and $2$, so this collection has no members at all. It is still perfectly well-defined — every object is decided, and the answer is always "no".

📘 Definition 1.3 — The empty set

A set which has no element is called the empty set (also the null set or void set). It is denoted by $\phi$ or by $\{\ \}$. There is only one empty set: any two sets with no elements have exactly the same elements (none), so they are equal.

📘 Definition 1.4 — Finite and infinite sets; $n(A)$

A set is finite if it is empty or if its elements can be counted, the counting coming to an end. A set that is not finite is infinite.

For a finite set $A$, the number of distinct elements of $A$ is written $n(A)$ (it is also called the cardinal number of $A$). Thus $n(\{C,O,M,I,T,E\})=6$ and $n(\phi)=0$.

Every finite set can in principle be written in roster form. An infinite set can be written in roster form only when its elements follow a clear pattern, like $\mathbb{N}=\{1,2,3,\dots\}$ or $\{\dots,-6,-3,0,3,6,\dots\}$. Many infinite sets — the real numbers between $0$ and $1$, the points on a line, the circles through a given point — have no such list at all, and set-builder form is the only sensible description.

✏️ Example 3 — Solution sets, and an empty one

Write in roster form: (a) $\{x : x\in\mathbb{R},\ x^2-5x+6=0\}$; (b) $\{x : x\in\mathbb{Z},\ 2x+1=0\}$; (c) $\{x : x\in\mathbb{R},\ x^2=-4\}$; (d) $\left\{x : x=\dfrac{n}{n+1},\ n\in\mathbb{N},\ n\le4\right\}$.

💡 Show solution▶

(a) Factorise: $x^2-5x+6=(x-2)(x-3)=0$, so $x=2$ or $x=3$. Both are real: $\{2,3\}$.

(b) $2x+1=0$ gives $x=-\tfrac12$, which is not an integer. No element survives: the set is $\phi$. (In $\mathbb{Q}$ the same equation would have solution set $\left\{-\tfrac12\right\}$ — the home set matters.)

(d) Put $n=1,2,3,4$: $\left\{\tfrac12,\tfrac23,\tfrac34,\tfrac45\right\}$.

Check (a) $2^2-5(2)+6=0$ and $3^2-5(3)+6=0$. ✓

✏️ Example 4 — Finite or infinite?

State whether each set is finite or infinite, and find $n(A)$ for the finite ones. (a) $A=\{x : x\in\mathbb{N},\ x \text{ is a multiple of } 7,\ x\lt100\}$; (b) $\{x : x\in\mathbb{Z},\ x\lt3\}$; (c) the months of the year that have $31$ days; (d) $\{x : x\in\mathbb{R},\ 0\lt x\lt1\}$; (e) $\{x : x\in\mathbb{N},\ x^2\lt50\}$.

💡 Show solution▶

(a) The multiples are $7,14,\dots,98$, i.e. $7k$ with $7k\lt100$, so $k\le14$ (since $7\times14=98$ and $7\times15=105$). Finite, $n(A)=14$.

(b) $\{\dots,-2,-1,0,1,2\}$ goes on for ever to the left: infinite.

(d) Between $0$ and $1$ lie $\tfrac12,\tfrac13,\tfrac14,\dots$ — already infinitely many. Infinite.

(e) $7^2=49\lt50$ but $8^2=64\gt50$, so the set is $\{1,2,3,4,5,6,7\}$: finite, $n=7$.

✏️ Example 5 — An infinite recipe can make a finite set

Write in roster form and find the number of elements: (a) $A=\{x : x=(-1)^n,\ n\in\mathbb{N}\}$; (b) $B=\{x : x=n^2-6n,\ n\in\mathbb{N},\ n\le6\}$.

💡 Show solution▶

(a) $n=1,2,3,4,\dots$ give $-1,1,-1,1,\dots$ Infinitely many values of $n$, but only two different outputs: $A=\{-1,1\}$, $n(A)=2$.

(b) $n=1,\dots,6$ give $-5,-8,-9,-8,-5,0$. The values $-5$ and $-8$ each appear twice, so $B=\{-9,-8,-5,0\}$ and $n(B)=4$, not $6$.

Lesson $n(A)$ counts distinct elements. Always remove repeats before you count — this is a favourite trap in objective tests.

Equal Sets

When are two sets "the same"? Not when they are described in the same words, nor when they are listed in the same order, but when they have exactly the same members.

📘 Definition 1.5 — Equal sets

Two sets $A$ and $B$ are equal, written $A=B$, if they have exactly the same elements: every element of $A$ is an element of $B$, and every element of $B$ is an element of $A$. Otherwise $A\ne B$.

✏️ Example 6 — Equal or not?

Decide whether $A=B$. (a) $A$ = the set of letters of "FOLLOW", $B$ = the set of letters of "WOLF". (b) $A=\{x : x\in\mathbb{N},\ x\lt5\}$, $B=\{0,1,2,3,4\}$. (c) $A=\{x : x\in\mathbb{R},\ x^2=9\}$, $B=\{x : x\in\mathbb{Z},\ |x|=3\}$. (d) $A=\{2,4,6\}$, $B=\{x : x \text{ is an even natural number}\}$.

💡 Show solution▶

(a) FOLLOW gives $\{F,O,L,W\}$; WOLF gives $\{W,O,L,F\}$. Same four elements in a different order: $A=B$.

(b) $A=\{1,2,3,4\}$ because $0\notin\mathbb{N}$. Since $0\in B$ but $0\notin A$, $A\ne B$.

(c) $x^2=9$ gives $x=\pm3$, so $A=\{-3,3\}$; $|x|=3$ also gives $B=\{-3,3\}$. $A=B$ — two quite different descriptions of one set.

(d) $8\in B$ but $8\notin A$, so $A\ne B$. One element that is in one set and not in the other is enough to prove two sets unequal.

Each description first produces a raw list (with repeats). Repeated entries are struck out — they add nothing to a set — and the distinct elements drop into the set. Then every element looks for its partner on the other side: a green link means "found", a red ring means "missing". The sets are equal exactly when nothing is left unmatched.

Pair:

raw list lengths—

$n(A)$—

$n(B)$—

unmatched elements—

$A=B$?—

⚠️ Common mistakes with sets

1. $\{0\}$ is not empty, and neither is $\{\phi\}$. $\{0\}$ has one element, the number $0$; $\{\phi\}$ has one element, the empty set. Only $\phi=\{\ \}$ has no elements. Never write $\{\phi\}$ when you mean $\phi$.

2. $0\notin\mathbb{N}$. In this course (as in the CBSE syllabus) the natural numbers start at $1$. Some computer languages and some other books start counting at $0$ — read the definition in front of you.

3. Count distinct elements. $n(\{x : x \text{ is a letter of "BANANA"}\})=n(\{B,A,N\})=3$, not $6$.

4. Same number of elements does not mean equal. $\{1,2\}$ and $\{3,4\}$ both have two elements but share none; they are not equal.

Exercises

Sets and membership. In parts 1–4 decide whether the collection is a set, giving a reason. In parts 5–8 insert $\in$ or $\notin$.

1 The vowels of the English alphabet.
2 The difficult chapters of the Class 11 mathematics syllabus.
3 The integers strictly between $-4$ and $4$.
4 The most beautiful cities of India.
5 With $A=\{1,3,5,7,9\}$: $\ 3\ \square\ A$, $\ 4\ \square\ A$, $\ 9\ \square\ A$.
6 $\sqrt2\ \square\ \mathbb{Q}$, $\ -7\ \square\ \mathbb{N}$, $\ 0.25\ \square\ \mathbb{Q}$, $\ \pi\ \square\ \mathbb{R}$.
7 $0\ \square\ \mathbb{N}$, $\ 0\ \square\ \mathbb{Z}$, $\ -\tfrac34\ \square\ \mathbb{Q}^+$, $\ 5\ \square\ \mathbb{Z}^+$.
8 With $P=\{x : x \text{ is a prime number}\}$: $\ 1\ \square\ P$, $\ 2\ \square\ P$, $\ 91\ \square\ P$, $\ 97\ \square\ P$.

💡 Show solutions▶

1 A set: membership is decided exactly; it is $\{a,e,i,o,u\}$.

2 Not a set: "difficult" differs from student to student.

3 A set: $\{-3,-2,-1,0,1,2,3\}$.

4 Not a set: "most beautiful" is a matter of opinion.

5 $3\in A$, $\ 4\notin A$, $\ 9\in A$.

6 $\sqrt2\notin\mathbb{Q}$ (irrational), $\ -7\notin\mathbb{N}$, $\ 0.25=\tfrac14\in\mathbb{Q}$, $\ \pi\in\mathbb{R}$.

7 $0\notin\mathbb{N}$, $\ 0\in\mathbb{Z}$, $\ -\tfrac34\notin\mathbb{Q}^+$ (it is negative), $\ 5\in\mathbb{Z}^+$.

8 $1\notin P$ (a prime has exactly two divisors; $1$ has one), $\ 2\in P$, $\ 91\notin P$ (because $91=7\times13$), $\ 97\in P$ (no prime up to $\sqrt{97}\approx9.8$ — that is $2,3,5,7$ — divides it).

Write each set in roster form.

1 $\{x : x\in\mathbb{Z},\ -2\lt x\le3\}$
2 $\{x : x \text{ is a prime number},\ x\lt20\}$
3 $\{x : x \text{ is a letter of the word "ALGEBRA"}\}$
4 $\{x : x\in\mathbb{R},\ x^2-7x+12=0\}$
5 $\{x : x\in\mathbb{N},\ x \text{ divides } 36\}$
6 $\{x : x=2n-1,\ n\in\mathbb{N},\ n\le6\}$
7 $\{x : x\in\mathbb{Z},\ x^2\le10\}$
8 $\left\{x : x=\dfrac{n}{n+2},\ n\in\mathbb{N},\ n\le3\right\}$
9 $\{x : x\in\mathbb{N},\ 3x+2\lt20\}$

Check: let the set-builder machine generate the roster for a part (work it out by hand first!)

Part:

candidates tested—

roster form—

number of elements—

💡 Show solutions▶

1 $-2$ is excluded, $3$ included: $\{-1,0,1,2,3\}$.

2 $\{2,3,5,7,11,13,17,19\}$.

3 A, L, G, E, B, R, A — the second A is a repeat: $\{A,L,G,E,B,R\}$.

4 $x^2-7x+12=(x-3)(x-4)$: $\{3,4\}$.

5 Pairs $1\times36,\ 2\times18,\ 3\times12,\ 4\times9,\ 6\times6$: $\{1,2,3,4,6,9,12,18,36\}$ (nine elements; $6$ is written once).

6 $n=1,\dots,6$: $\{1,3,5,7,9,11\}$.

7 $3^2=9\le10$ but $4^2=16\gt10$, and negatives behave the same way: $\{-3,-2,-1,0,1,2,3\}$.

8 $n=1,2,3$ give $\tfrac13,\ \tfrac24=\tfrac12,\ \tfrac35$: $\left\{\tfrac13,\tfrac12,\tfrac35\right\}$.

9 $3x\lt18$, so $x\lt6$: $\{1,2,3,4,5\}$.

III

Write each set in set-builder form.

1 $\{2,4,6,8,10\}$
2 $\{1,8,27,64,125\}$
3 $\{5,10,15,20,\dots\}$
4 $\{-4,-2,0,2,4\}$
5 $\left\{\tfrac12,\tfrac23,\tfrac34,\tfrac45,\tfrac56,\tfrac67\right\}$
6 $\left\{1,\tfrac12,\tfrac13,\tfrac14,\dots\right\}$
7 $\{11,13,17,19\}$

💡 Show solutions▶

1 $\{x : x=2n,\ n\in\mathbb{N},\ n\le5\}$, or $\{x : x \text{ is an even natural number},\ x\le10\}$.

2 Cubes: $\{x : x=n^3,\ n\in\mathbb{N},\ n\le5\}$.

3 $\{x : x=5n,\ n\in\mathbb{N}\}$ — the multiples of $5$; no upper limit, the set is infinite.

4 $\{x : x=2n,\ n\in\mathbb{Z},\ -2\le n\le2\}$.

5 Numerator $n$, denominator $n+1$: $\left\{x : x=\dfrac{n}{n+1},\ n\in\mathbb{N},\ n\le6\right\}$.

6 $\left\{x : x=\dfrac1n,\ n\in\mathbb{N}\right\}$.

7 $\{x : x \text{ is a prime number},\ 10\lt x\lt20\}$.

Empty, finite, infinite and equal sets.

1 Is $\{x : x\in\mathbb{N},\ x\lt1\}$ empty?
2 Is $\{x : x\in\mathbb{R},\ x^2+1=0\}$ empty?
3 Is $\{x : x \text{ is an even prime number}\}$ empty? If not, find its number of elements.
4 Finite or infinite: $\{x : x\in\mathbb{N},\ x \text{ is a multiple of } 5\}$?
5 Finite or infinite: $\{x : x\in\mathbb{Z},\ -50\le x\le50\}$? If finite, find $n$.
6 Is $\{1,2,3\}=\{x : x\in\mathbb{N},\ x^2\lt12\}$?
7 Is the set of letters of "LISTEN" equal to the set of letters of "SILENT"?
8 Let $A=\{0\}$, $B=\phi$, $C=\{\phi\}$. Find $n(A)$, $n(B)$, $n(C)$. Are any two of them equal?
9 Payments. A tea stall records the amounts (in ₹) of its UPI payments in one hour: $20,\ 45,\ 20,\ 300,\ 45,\ 80,\ 20$. Write the set $A$ of distinct amounts in roster form, find $n(A)$, and explain why $n(A)$ is not $7$.
10 Cricket. The runs scored off the six balls of an over were $1,4,0,6,1,1$. Write the set $R$ of different scores, find $n(R)$, and decide whether $2\in R$.
11 Show that $\{x : x=3n+2,\ n\in\mathbb{Z}\}=\{x : x=3n-1,\ n\in\mathbb{Z}\}$.
12 Challenge. Find $n(S)$, where $S=\left\{n\in\mathbb{Z} : \dfrac{n+5}{n+1}\in\mathbb{Z}\right\}$.

💡 Show solutions▶

1 Yes: the smallest natural number is $1$, so none is less than $1$. The set is $\phi$.

2 Yes: $x^2+1\ge1\gt0$ for every real $x$, so the set is $\phi$.

3 Not empty: $2$ is prime and even, and every other even number is divisible by $2$ and so not prime. The set is $\{2\}$, with $1$ element.

4 Infinite: $\{5,10,15,\dots\}$ never ends.

5 Finite: $50$ negative integers, $50$ positive integers and $0$, so $n=101$.

6 Yes: $1,4,9\lt12$ but $16\gt12$, so the right-hand set is $\{1,2,3\}$.

7 Yes: both are $\{L,I,S,T,E,N\}$ (the two words are anagrams).

8 $n(A)=1$, $n(B)=0$, $n(C)=1$. $B$ differs from both others since it has no elements. $A\ne C$: the only element of $A$ is the number $0$, the only element of $C$ is the set $\phi$. No two are equal.

9 $A=\{20,45,80,300\}$ (amounts in ₹), $n(A)=4$. There were $7$ payments, but a set records only which amounts occurred, not how often: $20$ appeared three times and $45$ twice.

10 $R=\{0,1,4,6\}$, $n(R)=4$, and $2\notin R$.

11 If $x=3n+2$ then $x=3(n+1)-1$, which has the second form with the integer $n+1$. If $x=3m-1$ then $x=3(m-1)+2$, which has the first form with the integer $m-1$. Each set's elements lie in the other, so the sets are equal (both are the integers that leave remainder $2$ on division by $3$).

12 $\dfrac{n+5}{n+1}=1+\dfrac{4}{n+1}$, which is an integer exactly when $n+1$ divides $4$: $n+1\in\{\pm1,\pm2,\pm4\}$, so $n\in\{0,-2,1,-3,3,-5\}$ and $n(S)=6$. (Forgetting the negative divisors gives the wrong answer $3$.)

🎯 Exam corner Board · 1 markBoard · 2 marksJEE MainJEE Advanced

How this is examined: in the Board paper this section gives quick 1-mark MCQs and Assertion–Reason items and 2-mark roster/set-builder conversions; JEE problems hide the set idea inside a counting or divisibility question, and the usual trap is counting repeated or excluded elements.

E1. Board · 1 mark Which of the following is the empty set? (A) $\{x : x\in\mathbb{N},\ 1\lt x\lt2\}$ (B) $\{x : x\in\mathbb{Z},\ x^2=4\}$ (C) $\{0\}$ (D) $\{x : x \text{ is an even prime}\}$

💡 Show solution▶

Idea Find the elements of each set. (A): no natural number lies strictly between $1$ and $2$ — empty. (B) $=\{-2,2\}$. (C) has the element $0$. (D) $=\{2\}$.

Trap (C) looks "empty" because $0$ means "nothing" — but $\{0\}$ contains one element, the number $0$.

Answer (A).

E2. Board · 2 marks (i) Write $\{x : x\in\mathbb{Z},\ x^2\lt20\}$ in roster form. (ii) Write $\{1,4,9,16,25,36\}$ in set-builder form.

💡 Show solution▶

(i) $4^2=16\lt20$ and $5^2=25\gt20$, and $(-x)^2=x^2$: $\{-4,-3,-2,-1,0,1,2,3,4\}$. (1 mark)

(ii) $\{x : x=n^2,\ n\in\mathbb{N},\ 1\le n\le6\}$. (1 mark)

Answer (i) nine elements, from $-4$ to $4$; (ii) as above.

E3. Board · 1 mark Assertion (A): The set of letters of the word "BANANA" has $3$ elements. Reason (R): In a set, the order in which the elements are listed matters. Choose: (A) both A and R are true and R is the correct explanation of A; (B) both are true but R is not the correct explanation of A; (C) A is true but R is false; (D) A is false but R is true.

💡 Show solution▶

Idea The letters give $\{B,A,N\}$, so A is true. Order never matters in a set, so R is false. (The reason A is true is that repetitions are ignored — a different rule.)

Answer (C).

E4. JEE Main Numerical value. Find the number of elements of $\{x\in\mathbb{N} : x^2+3x \text{ is a two-digit number}\}$.

💡 Show solution▶

Idea $x^2+3x$ increases with $x$ on $\mathbb{N}$, so the members form a block of consecutive natural numbers. We need $10\le x^2+3x\le99$.

Ends $x=1$: $4$ (too small); $x=2$: $10$ ✓. $x=8$: $88$ ✓; $x=9$: $108$ (too big).

Answer $x\in\{2,3,\dots,8\}$: $7$ elements.

E5. JEE Main Let $S=\left\{n\in\mathbb{Z} : \dfrac{2n+3}{n-1}\in\mathbb{Z}\right\}$. Then $n(S)$ equals (A) $2$ (B) $3$ (C) $4$ (D) infinitely many

💡 Show solution▶

Idea Divide: $\dfrac{2n+3}{n-1}=2+\dfrac{5}{n-1}$. This is an integer exactly when $n-1$ divides $5$.

Divisors $n-1\in\{1,-1,5,-5\}$, so $n\in\{2,0,6,-4\}$. Check $n=-4$: $\dfrac{-5}{-5}=1$ ✓.

Trap Using only the positive divisors $1,5$ gives (A).

Answer (C). ⏱ Fast route: the number of integer divisors of $5$ (positive and negative) is $2\times2=4$.

E6. JEE Advanced One or more options correct. Let $A=\{x : x=4k+3,\ k\in\mathbb{Z}\}$, $B=\{x : x=4k-1,\ k\in\mathbb{Z}\}$, $C=\{x : x=2k+1,\ k\in\mathbb{Z}\}$. Which of the following are true? (A) $A=B$ (B) $A=C$ (C) $11\in B$ (D) $-5\in A$

💡 Show solution▶

(A) $4k+3=4(k+1)-1$ and $4k-1=4(k-1)+3$, so each set lies in the other: $A=B$. True.

(B) $1\in C$ ($k=0$), but $1=4k+3$ needs $k=-\tfrac12\notin\mathbb{Z}$, so $1\notin A$. False.

(C), (D) $11=4(3)-1$ ✓; $\ -5=4(-2)+3$ ✓. Both true.

Answer (A), (C), (D). A and B are both "the integers leaving remainder 3 on division by 4"; C is all odd integers, which is larger.

🔬 Research–Teaching NexusMini project · about 40 minutes

Research question. How does a search filter on a shopping or train-booking app turn your choices into a set — and when does it return the empty set?

Why it matters. Every filter ("price under ₹500", "4 stars and above", "departs after 18:00") is a property $P(x)$, and the results page is the set $\{x : P(x)\}$ in roster form. Databases are built on exactly this idea; the query language SQL even writes it as SELECT … WHERE condition. A results page that says "No items found" is showing you $\phi$.

What to do.

Make a small "catalogue" in a spreadsheet: $15$ items (phones, books or trains) with three columns such as price, rating and brand. Real data from any app is fine.
Write three filters in set-builder form, e.g. $A=\{x : \text{price}(x)\lt 15000\}$, $B=\{x : \text{rating}(x)\ge4.2\}$, and one combined filter.
Use the spreadsheet's filter tool to list each set in roster form and record $n(A)$, $n(B)$.
Find a filter combination that returns the empty set, and two differently worded filters that return equal sets.

Tools: Spreadsheet (filter or COUNTIF), or pen and paper with a printed list

Hand in: A one-page table of your filters in set-builder form, their roster forms and $n$-values, with one sentence explaining why two different descriptions gave the same set.

Going further. Combining filters with "and" and "or" is the intersection and union of sets — Section 1.3. Predict the size of the "and" result from your table before you check it.

Instructor note: look for correct use of $\lt$ versus $\le$ at price boundaries (a ₹15,000 phone in or out?) — exactly the end-point issue of Example 2 — and for students who count repeated brands as separate elements.

🧪 Self-check quiz

0/5 correct

Q1. Which of the following is a set?

Q2. The set of letters of the word "ASSESS" in roster form is

Q3. $\{x : x\in\mathbb{N},\ x\lt1\}$ is

Q4. If $A=\{x : x\in\mathbb{Z},\ -2\le x\lt3\}$, then $n(A)$ is

Q5. Which pair of sets is equal?

← Chapter 1 — Sets 1.2 Subsets, Intervals and the Universal Set →