Subsets

Written by TopicLamp Editorial Team Checked by TopicLamp Review Team Updated 8 Nov 2026 15 min read

NCERT Class 11 Mathematics Chapter 1 Section 1.6

Quick answer

A is a subset of B, written A ⊂ B, when every element of A is also in B; every set is a subset of itself, and the empty set is a subset of every set. A ⊂ B with A ≠ B makes A a proper subset of B. For a < b, intervals are subsets of R: (a, b) leaves out both endpoints, [a, b] includes both, and the half-open intervals include one; each has length b − a.

With your textbook

NCERT Mathematics (Class 11), Chapter 1 “Sets”, Section 1.6 “Subsets”, pages 9–12.

Open the official NCERT chapter (PDF, 2.2 MB, ncert.nic.in) (opens in a new tab)

Checked against the 2026-27 reprint on 19 Oct 2026.

What is a subset?

Subset

A set A is said to be a subset of a set B if every element of A is also an element of B.

NCERT Mathematics (Class 11), Section 1.6, page 9

In symbols, A ⊂ B if a ∈ A ⇒ a ∈ B, read “A is a subset of B” or “A is contained in B”; if not, A ⊄ B.
A ⊂ B and B ⊂ A together mean A = B (NCERT page 9).
Every set is a subset of itself, A ⊂ A, and the empty set ϕ is a subset of every set.

In plain words: A is a subset of B when A lies wholly inside B, with nothing of A left outside. The students of your class form a subset of the students of your school (NCERT page 9). A subset can be all of B (B ⊂ B) or none of it (ϕ ⊂ B); anything in between counts too.

Proper subsets, supersets and singletons

Let A and B be two sets. If A ⊂ B and A ≠ B , then A is called a proper subset of B and B is called superset of A.

NCERT Mathematics (Class 11), Section 1.6, page 10

For example, A = {1, 2, 3} is a proper subset of B = {1, 2, 3, 4}. A set with only one element, such as {a}, is a singleton set (NCERT page 10).

NCERT’s other examples (page 10):

the rational numbers form a subset of the real numbers, Q ⊂ R;
if A is the set of divisors of 56 and B the set of its prime divisors, then B ⊂ A: B = {2, 7}, A = {1, 2, 4, 7, 8, 14, 28, 56};
{1, 3, 5} and {x : x is an odd natural number less than 6} are each a subset of the other, so they are equal;
{a, e, i, o, u} and {a, b, c, d}: neither is a subset of the other, since e is not in the second and b is not in the first (NCERT Example 10).

Outside mathematics too: the seven SI base units form a subset of the set of all SI units, which also contains the derived units such as the newton. In biology, the organisms of a species form a subset of the organisms of its genus, and those of a genus a subset of those of its family, up the taxonomic hierarchy.

A note on symbols. NCERT writes ⊂ for every subset, including A ⊂ A. Many books and exam papers write ⊆ for “subset, possibly equal” and keep ⊂ for a proper subset. Read which convention a question uses; in NCERT, ⊂ allows equality.

∈ and ⊂ are different

∈ joins an element to a set; ⊂ joins a set to a set. For A = {1, 2, {3, 4}, 5}: 1 ∈ A and {1, 2, 5} ⊂ A, but writing 1 ⊂ A is wrong, because 1 is not a set. {3, 4} ∈ A, since {3, 4} is one of the elements listed; {3, 4} ⊄ A, since 3 and 4 themselves are not elements of A; and {{3, 4}} ⊂ A.

NCERT Example 11 (page 10) shows that A ∈ B and B ⊂ C do not give A ⊂ C: with A = {1}, B = {{1}, 2} and C = {{1}, 2, 3}, A ∈ B and B ⊂ C, but 1 ∈ A while 1 ∉ C, so A ⊄ C. By contrast, ⊂ is transitive: if A ⊂ B and B ⊂ C, then A ⊂ C, since every element of A is in B and so in C.

Subsets of the real numbers

Some important subsets of R (NCERT Section 1.6.1, pages 10–11):

natural numbers N = {1, 2, 3, 4, 5, …};
integers Z = {…, −3, −2, −1, 0, 1, 2, 3, …};
rational numbers Q = {x : x = p/q, p, q ∈ Z and q ≠ 0}, such as −5 = −5/1, 5/7, 3½ = 7/2 and −11/3;
irrational numbers, which NCERT writes T = {x : x ∈ R and x ∉ Q}, such as √2, √5 and π.

They are related by N ⊂ Z ⊂ Q, Q ⊂ R, T ⊂ R and N ⊄ T (NCERT page 11). In fact Q and T share no element, and together they make up R.

Nested regions: N inside Z inside Q, all inside R; the irrational numbers T fill the rest of R, outside Q. Examples: 5 in N, −3 in Z, 5/7 in Q, square root of 2 and pi in T. — N ⊂ Z ⊂ Q ⊂ R; the irrationals T are the part of R outside Q (NCERT Section 1.6.1, pages 10–11).

Intervals as subsets of R

Let a, b ∈ R with a < b (NCERT Section 1.6.2, page 11):

Interval	Set-builder form	Endpoints
(a, b)	{x : a < x < b}	neither included (open)
[a, b]	{x : a ≤ x ≤ b}	both included (closed)
[a, b)	{x : a ≤ x < b}	a only
(a, b]	{x : a < x ≤ b}	b only

A round bracket leaves the endpoint out; a square bracket puts it in. NCERT page 11 introduces the last two as “closed at one end and open at the other” but then calls each an “open interval” from a to b; they are usually called half-open (or semi-open) intervals, since one end is included and the other is not.

Intervals can also run to infinity: [0, ∞) is the non-negative real numbers, (−∞, 0) the negative real numbers and (−∞, ∞) the whole of R. Infinity is not a number, so ∞ always takes a round bracket. Domains and ranges of real functions are often intervals like these: f(x) = x² has range [0, ∞), and sin x has domain (−∞, ∞) and range [−1, 1].

Here, we note that an interval contains infinitely many points.

NCERT Mathematics (Class 11), Section 1.6.2, page 11

The number b − a is called the length of each of these four intervals (NCERT page 12).

For example, (−3, 5) ⊂ [−7, 9] (NCERT page 11); {x : x ∈ R, −5 < x ≤ 7} = (−5, 7]; and [−3, 5) = {x : −3 ≤ x < 5}. Solution sets of inequalities are written the same way: 5x − 3 < 3x + 1 gives x < 2, the interval (−∞, 2).

Four intervals from a to b on number lines: (a, b) hollow at both ends, [a, b] filled at both, [a, b) filled at a only, (a, b] filled at b only. Each has length b minus a. — Filled dots are included and hollow dots are left out; all four intervals have length b − a (after NCERT Fig. 1.1, page 11).

Solved examples

Example 1: ⊂ or ⊄? (NCERT Example 9, page 10)

For ϕ, A = {1, 3}, B = {1, 5, 9} and C = {1, 3, 5, 7, 9}, fill in ⊂ or ⊄: (i) ϕ … B; (ii) A … B; (iii) A … C; (iv) B … C.

(i) The empty set is a subset of every set. (ii) 3 ∈ A but 3 ∉ B. (iii) 1 and 3 are both in C. (iv) 1, 5 and 9 are all in C.

Answer: (i) ϕ ⊂ B; (ii) A ⊄ B; (iii) A ⊂ C; (iv) B ⊂ C.

Example 2: Element or subset? (NCERT Exercise 1.3, Q3)

For A = {1, 2, {3, 4}, 5}, which are incorrect: (i) {3, 4} ⊂ A; (ii) {3, 4} ∈ A; (iii) {{3, 4}} ⊂ A; (iv) 1 ∈ A; (v) 1 ⊂ A; (vi) {1, 2, 5} ⊂ A; (vii) {1, 2, 5} ∈ A; (viii) {1, 2, 3} ⊂ A; (ix) ϕ ∈ A; (x) ϕ ⊂ A; (xi) {ϕ} ⊂ A?

The elements of A are 1, 2, {3, 4} and 5. (i) 3 ∉ A. (v) 1 is not a set. (vii) {1, 2, 5} is not one of the four elements. (viii) 3 ∉ A. (ix) ϕ is not listed as an element. (xi) {ϕ} ⊂ A would need ϕ ∈ A.

Answer: incorrect: (i), (v), (vii), (viii), (ix) and (xi). Correct: (ii), (iii), (iv), (vi) and (x).

Example 3: All subsets (NCERT Exercise 1.3, Q4)

Write all the subsets of (i) {a}; (ii) {a, b}; (iii) {1, 2, 3}; (iv) ϕ.

List by size: none, one element, two elements, and so on.

Answer: (i) ϕ, {a}; (ii) ϕ, {a}, {b}, {a, b}; (iii) ϕ, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}; (iv) ϕ only. The counts are 2, 4, 8 and 1, that is 2¹, 2², 2³ and 2⁰: a set with n elements has 2ⁿ subsets (together they form its power set; see why there are 2ⁿ).

Example 4: Intervals (NCERT Exercise 1.3, Q5 and Q6)

(a) Write as intervals: {x : x ∈ R, −4 < x ≤ 6} and {x : x ∈ R, 0 ≤ x < 7}. (b) Write in set-builder form: (6, 12] and [−23, 5).

A strict inequality gives a round bracket; ≤ gives a square one.

Answer: (a) (−4, 6] and [0, 7); (b) {x : x ∈ R, 6 < x ≤ 12} and {x : x ∈ R, −23 ≤ x < 5}.

Common mistakes to avoid

Common mistake

Mixing up ∈ and ⊂

∈ is used between an element and a set, ⊂ between two sets: 2 ∈ {1, 2} but {2} ⊂ {1, 2}. Writing 2 ⊂ {1, 2} is wrong. Watch the empty set too: ϕ ∈ {ϕ} and ϕ ⊂ {ϕ} are both true, but ϕ ∈ {1, 2} is false.

Common mistake

Forgetting ϕ and the set itself

Every set has ϕ and itself among its subsets. {1, 2, 3} has 8 subsets, not 6.

Common mistake

Getting brackets wrong at infinity

(−∞, 0) and [0, ∞) are right; [0, ∞] is wrong, because ∞ is not a real number and cannot be included.

Common mistake

Calling [a, b) an open interval

Only (a, b) is open and only [a, b] is closed; [a, b) includes a, so it is half-open, not open.

Quick revision summary

A ⊂ B: every element of A is in B. A ⊂ B and B ⊂ A ⇔ A = B. ϕ ⊂ A and A ⊂ A for every A.
Proper subset: A ⊂ B and A ≠ B; then B is a superset of A. A singleton has one element.
∈ links an element to a set; ⊂ links two sets.
N ⊂ Z ⊂ Q ⊂ R; irrationals T = {x : x ∈ R and x ∉ Q}.
Intervals: (a, b) open; [a, b] closed; half-open [a, b) with a ≤ x < b, and (a, b] with a < x ≤ b; length b − a; ∞ always gets a round bracket.

Formula sheet

A ⊂ B ⇔ (a ∈ A ⇒ a ∈ B); A ⊂ B and B ⊂ A ⇔ A = B
(a, b) = {x : a < x < b}; [a, b] = {x : a ≤ x ≤ b}
[a, b) = {x : a ≤ x < b}; (a, b] = {x : a < x ≤ b}; length b − a
A set with n elements has 2ⁿ subsets

Questions students ask

What is the difference between a subset and a proper subset?

A subset of B may be the whole of B, while a proper subset of B must leave out at least one element of B: A ⊂ B means that every element of A is in B, and A is a proper subset of B when, in addition, A ≠ B. So {2, 4} and {2, 4, 8} are both subsets of {2, 4, 8}, but only {2, 4} is a proper subset of it. Every set is a subset of itself but never a proper subset of itself, and a set with n elements has 2ⁿ − 1 proper subsets.

What is an improper subset of a set?

The improper subset of a set A is A itself, the one subset of A that is not a proper subset, since A ⊂ A holds but A ≠ A does not. Some books also call the empty set ϕ an improper subset and keep the word proper for the subsets that are neither ϕ nor A; under that convention a non-empty set with n elements has 2ⁿ − 2 proper subsets, while NCERT’s definition (A ⊂ B and A ≠ B) makes ϕ a proper subset of every non-empty set and gives 2ⁿ − 1. Check which convention a question follows before counting.

What are the symbols for subset, proper subset and superset?

The subset symbol in NCERT is ⊂: A ⊂ B means “A is a subset of B” and allows A = B, A ⊄ B means that A is not a subset of B, and a proper subset is written A ⊂ B together with A ≠ B. Many other books and exam papers write A ⊆ B for a subset that may equal B and keep A ⊂ B for a proper subset. The superset symbol faces the other way: B ⊃ A, read “B contains A”, says the same as A ⊂ B, and books that write ⊆ write ⊇ in the same way; NCERT calls B a superset of A when A is a proper subset of B.

Why is the empty set a subset of every set?

The empty set ϕ is a subset of every set A because the test for ϕ ⊂ A, that every element of ϕ is also an element of A, cannot fail: ϕ has no elements, so none of them can be missing from A. To show ϕ ⊄ A you would need an element of ϕ that is not in A, and there is none. So ϕ ⊂ A for every set A, ϕ ⊂ ϕ included, and ϕ is a proper subset of every set except ϕ itself.

Can a set be both an element and a subset of another set?

Yes, a set can be both an element and a subset of another set: in B = {a, {a}}, the set {a} is an element because it is listed, and a subset because its only element a is listed too, so {a} ∈ B and {a} ⊂ B. Being an element does not make it a subset: for distinct ordinary objects a and b, in C = {b, {a}}, {a} ∈ C but {a} ⊄ C, since a itself is not an element of C. Test ∈ by finding the object in the list, and ⊂ by finding each of its elements there.

Practice questions

Try each question first, then open the answer to check your working.

Q1. Fill in ⊂ or ⊄: (i) {2, 3, 4} … {1, 2, 3, 4, 5}; (ii) {a, b, c} … {b, c, d}; (iii) {x : x is an equilateral triangle in a plane} … {x : x is a triangle in the same plane}; (iv) {x : x is an even natural number} … {x : x is an integer} (NCERT Exercise 1.3, Q1 (i), (ii), (vi), (vii)).

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(i) ⊂; (ii) ⊄, since a is not in the second set; (iii) ⊂; (iv) ⊂.
Q2. True or false: (i) {a, b} ⊄ {b, c, a}; (ii) {a, e} ⊂ {x : x is a vowel}; (iii) {1, 2, 3} ⊂ {1, 3, 5}; (iv) {a} ∈ {a, b, c} (NCERT Exercise 1.3, Q2 (i)–(iii), (v)).

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(i) False: {a, b} ⊂ {b, c, a}. (ii) True. (iii) False: 2 is not in {1, 3, 5}. (iv) False: {a} is a subset, not an element; a ∈ {a, b, c}.
Q3. Write as intervals: (i) {x : x ∈ R, −12 < x < −10}; (ii) {x : x ∈ R, 3 ≤ x ≤ 4} (NCERT Exercise 1.3, Q5 (ii), (iv)).

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(i) (−12, −10); (ii) [3, 4].
Q4. Write (−3, 0) and [6, 12] in set-builder form (NCERT Exercise 1.3, Q6 (i), (ii)).

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{x : x ∈ R, −3 < x < 0} and {x : x ∈ R, 6 ≤ x ≤ 12}.
Q5. How many subsets does {p, q, r, s} have, and how many of them are proper subsets?

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2⁴ = 16 subsets; all except the set itself are proper, so 15.
Q6. What is the length of the interval [−2, 5)? Does it contain −2? Does it contain 5?

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Length 5 − (−2) = 7. It contains −2 (square bracket) but not 5 (round bracket).

Subsets

What is a subset?

Proper subsets, supersets and singletons

∈ and ⊂ are different

Subsets of the real numbers

Intervals as subsets of R

Solved examples

Example 1: ⊂ or ⊄? (NCERT Example 9, page 10)

Example 2: Element or subset? (NCERT Exercise 1.3, Q3)

Example 3: All subsets (NCERT Exercise 1.3, Q4)

Example 4: Intervals (NCERT Exercise 1.3, Q5 and Q6)

Common mistakes to avoid

Quick revision summary

Questions students ask

What is the difference between a subset and a proper subset?

What is an improper subset of a set?

What are the symbols for subset, proper subset and superset?

Why is the empty set a subset of every set?

Can a set be both an element and a subset of another set?

Practice questions

Sources and review

Topics in Sets