NCERT Class 11 Mathematics — Sets, Exercise 1.1. All 6 questions solved.

This is the first exercise of the chapter and it does only two things, but it does them repeatedly: it asks whether a given collection is a set at all, and it asks you to move between the two ways of writing one down.

• Roster form lists the elements between braces: $\{1, 2, 3, 4, 5, 6\}$. Order does not matter, and no element is written twice.
• Set-builder form states the defining property: ${x : x \text{ is a natural number less than } 7}$. Read the colon as “such that”.
• $a \in A$ means $a$ belongs to $A$; $a \notin A$ means it does not.

Key insight. A collection is a set only when membership is decidable without argument — given any object, everyone must reach the same yes-or-no. “All even integers” qualifies; “the ten most talented writers of India” does not, because talent is a matter of opinion and two people would produce two different lists. This one test settles every part of question 1.

Question 1

Which of the following are sets? Justify your answer.

(i) The collection of all the months of a year beginning with the letter J. (ii) The collection of ten most talented writers of India. (iii) A team of eleven best-cricket batsmen of the world. (iv) The collection of all boys in your class. (v) The collection of all natural numbers less than 100. (vi) A collection of novels written by the writer Munshi Prem Chand. (vii) The collection of all even integers. (viii) The collection of questions in this Chapter. (ix) A collection of most dangerous animals of the world.

Solution. Apply the test above to each collection in turn — is membership decidable, or does it rest on someone’s judgement?

(i) A set. The months are fixed and public knowledge, so the collection is exactly $\{\text{January}, \text{June}, \text{July}\}$. There is nothing to argue about.

(ii) Not a set. “Most talented” has no agreed measure. Two readers would name two different tens, so no object has a definite membership status.

(iii) Not a set. Same defect — “best” is a matter of opinion, and the eleven names would differ from selector to selector.

(iv) A set. The boys in your class form a definite, countable list; anyone looking at the register would produce the same one.

(v) A set. The natural numbers below $100$ are precisely $1, 2, \ldots, 99$.

(vi) A set. Munshi Prem Chand’s novels are a fixed, documented body of work.

(vii) A set. An integer either is divisible by $2$ or is not; the test is exact.

(viii) A set. The questions printed in this chapter are a definite collection — you can point at each one.

(ix) Not a set. “Most dangerous” is a judgement, not a measurable property.

Question 2

Let $A = \{1, 2, 3, 4, 5, 6\}$. Insert the appropriate symbol $\in$ or $\notin$ in the blank spaces:

(i) $5 \ldots A$    (ii) $8 \ldots A$    (iii) $0 \ldots A$ (iv) $4 \ldots A$    (v) $2 \ldots A$    (vi) $10 \ldots A$

Solution. Read each number off the roster and ask whether it appears in the list. $A$ contains exactly $1, 2, 3, 4, 5$ and $6$ — nothing else.

A note on the printed key. NCERT’s answer key labels the six parts “(i) (ii) (iii) (vi) (v) (vi)” — the fourth label should read (iv). Every symbol it gives is correct; only the numeral is misprinted, so match them in order rather than by label.

Question 3

Write the following sets in roster form:

(i) $A = \{x : x \text{ is an integer and } -3 \le x < 7\}$ (ii) $B = \{x : x \text{ is a natural number less than } 6\}$ (iii) $C = \{x : x \text{ is a two-digit natural number such that the sum of its digits is } 8\}$ (iv) $D = \{x : x \text{ is a prime number which is a divisor of } 60\}$ (v) $E = $ The set of all letters in the word TRIGONOMETRY (vi) $F = $ The set of all letters in the word BETTER

Solution. Roster form means listing every element once. Two details decide most of the marks here: whether the endpoints are included, and the rule that a repeated letter is written only once.

(i) The condition $-3 \le x < 7$ includes $-3$ but excludes $7$:

$$A = \{-3, -2, -1, 0, 1, 2, 3, 4, 5, 6\}$$

(ii) Natural numbers start at $1$, so “less than 6” gives

$$B = \{1, 2, 3, 4, 5\}$$

(iii) A two-digit number $\overline{ab}$ has digit sum $a + b = 8$ with $a \ge 1$. Running $a$ from $1$ to $8$ gives $17, 26, 35, 44, 53, 62, 71, 80$ (and $a = 9$ would force $b = -1$, which is impossible):

$$C = \{17, 26, 35, 44, 53, 62, 71, 80\}$$

(iv) Factorising, $60 = 2^2 \times 3 \times 5$, so its prime divisors are

$$D = \{2, 3, 5\}$$

(v) TRIGONOMETRY has twelve letters, but T, R and O each repeat. A set records each distinct letter once:

$$E = \{\mathrm{T, R, I, G, O, N, M, E, Y}\}$$

(vi) BETTER has six letters; E and T repeat:

$$F = \{\mathrm{B, E, T, R}\}$$

Question 4

Write the following sets in the set-builder form:

(i) $\{3, 6, 9, 12\}$    (ii) $\{2, 4, 8, 16, 32\}$    (iii) $\{5, 25, 125, 625\}$ (iv) $\{2, 4, 6, \ldots\}$    (v) $\{1, 4, 9, \ldots, 100\}$

Solution. The work is spotting the pattern and then bounding it, because a finite set needs a range on $n$ as well as a formula. Notice the difference in kind between (ii) and (iii) — both are geometric, but with different bases — and between (i) and (v), one arithmetic and one square.

(i) Each element is a multiple of $3$, from the first to the fourth:

$$\{x : x = 3n,\; n \in \mathbf{N} \text{ and } 1 \le n \le 4\}$$

(ii) Powers of $2$: $2^1, 2^2, 2^3, 2^4, 2^5$:

$$\{x : x = 2^n,\; n \in \mathbf{N} \text{ and } 1 \le n \le 5\}$$

(iii) Powers of $5$: $5^1, 5^2, 5^3, 5^4$:

$$\{x : x = 5^n,\; n \in \mathbf{N} \text{ and } 1 \le n \le 4\}$$

(iv) The dots signal that the list never ends, so no bound is imposed:

$$\{x : x \text{ is an even natural number}\}$$

(v) These are $1^2, 2^2, 3^2, \ldots, 10^2$, ending at $100$:

$$\{x : x = n^2,\; n \in \mathbf{N} \text{ and } 1 \le n \le 10\}$$

Question 5

List all the elements of the following sets:

(i) $A = \{x : x \text{ is an odd natural number}\}$ (ii) $B = \left\{x : x \text{ is an integer},\; -\dfrac{1}{2} < x < \dfrac{9}{2}\right\}$ (iii) $C = \{x : x \text{ is an integer},\; x^2 \le 4\}$ (iv) $D = \{x : x \text{ is a letter in the word “LOYAL”}\}$ (v) $E = \{x : x \text{ is a month of a year not having } 31 \text{ days}\}$ (vi) $F = \{x : x \text{ is a consonant in the English alphabet which precedes } k\}$

Solution.

(i) Infinitely many, so the list is left open:

$$A = \{1, 3, 5, 7, \ldots\}$$

(ii) The bound is $-0.5 < x < 4.5$, and $x$ must be an integer. The integers in that window are $0, 1, 2, 3, 4$ — note that $-\tfrac12$ excludes $-1$ but admits $0$:

$$B = \{0, 1, 2, 3, 4\}$$

(iii) $x^2 \le 4$ means $-2 \le x \le 2$, and both endpoints qualify since $(\pm 2)^2 = 4$:

$$C = \{-2, -1, 0, 1, 2\}$$

(iv) LOYAL repeats L, so it is listed once:

$$D = \{\mathrm{L, O, Y, A}\}$$

(v) The months with fewer than $31$ days are the four thirty-day months plus February:

$$E = \{\text{February},\, \text{April},\, \text{June},\, \text{September},\, \text{November}\}$$

(vi) The letters before $k$ are $a$ to $j$; strike out the vowels $a$, $e$, $i$:

$$F = \{b, c, d, f, g, h, j\}$$

Question 6

Match each of the sets on the left in the roster form with the same set on the right described in set-builder form:

Solution. Expand each description on the right into roster form and the matching becomes mechanical.

• (a) Divisors of $6$ are $1, 2, 3, 6$; of those, the primes are $2$ and $3$, so (a) $= \{2, 3\}$.
• (b) Odd naturals below $10$: $\{1, 3, 5, 7, 9\}$.
• (c) All natural divisors of $6$: $\{1, 2, 3, 6\}$.
• (d) MATHEMATICS has eleven letters, with M, A and T repeated; the distinct ones are $\{\mathrm{M, A, T, H, E, I, C, S}\}$.

The point of pairing (a) with (c) is the contrast: the same number $6$ gives two different sets depending on whether “prime” is imposed. Reading the condition to the end is the whole task.

Common mistakes

• Question 1(viii) read as “not a set”. Students see a vague-sounding phrase and reject it, but “the questions in this Chapter” is perfectly definite — you can count them. Vagueness of language is not the test; vagueness of membership is.
• Question 3(i), forgetting that $-3$ is included. The condition is $-3 \le x < 7$: the left endpoint carries $\le$ and belongs, the right carries $<$ and does not. Mixing these up costs two elements at once.
• Question 3(v) and (vi), repeating letters. Writing $E$ with two T’s or $F$ as $\{\mathrm{B, E, T, T, E, R}\}$ misses the defining feature of a set: an element is either in or out, never in twice.
• Question 4, giving a formula with no range. $\{x : x = 3n,\ n \in \mathbf{N}\}$ describes $\{3, 6, 9, 12, 15, \ldots\}$, an infinite set — not the four-element set asked for. A finite set needs the bound $1 \le n \le 4$.
• Question 5(ii), including $-1$ or leaving out $0$. The lower bound $-\tfrac12$ sits between $-1$ and $0$, so $0$ is the smallest integer admitted. Converting the fractions to $-0.5$ and $4.5$ before listing removes the guesswork.
• Question 5(vi), counting the vowels. The question says consonant, so $a$, $e$ and $i$ are excluded even though they precede $k$ — and $j$ is included even though it is easy to forget at the end of the run.
• Question 6, matching (i) with (a). $\{1, 2, 3, 6\}$ is every divisor of $6$, which is (c). Option (a) adds the word “prime” and shrinks the set to $\{2, 3\}$.

Practise next

• Exercise 1.2 — the empty set, and finite versus infinite sets. It builds directly on the roster work above: deciding whether a set is empty usually means trying to list it and failing.
• Exercise 1.3 — subsets, intervals and the power set, where set-builder form from question 4 becomes the standard way of describing intervals of $\mathbf{R}$.