MathsISC Class 11 · Sets

Subsets, the Power Set and Intervals: Counting and Operations

Q: What is the difference between ⊂ and ∈?

a ∈ A says a is an element of A; A ⊂ B says every element of A is in B. An element and a subset are different things.

Q: Is ∅ a proper subset of ∅? Is it a proper subset of {1}?

Not of ∅: ∅ ⊆ ∅ but they are equal. It is a proper subset of every non-empty set, so ∅ is a proper subset of {1}.

Updated 17 October 2026 · Fact-checked

A set with n elements has 2ⁿ subsets, 2ⁿ − 1 proper subsets and C(n, k) subsets with k elements. The power set P(A) is the set of all subsets of A. Intervals such as (a, b), [a, b] and (a, b] are subsets of ℝ and are combined with union, intersection and difference.

Understand Subsets, Power Set and Intervals

Subsets. A is a subset of B (A ⊆ B) if every element of A is in B. The empty set ∅ is a subset of every set, and every set is a subset of itself. A is a proper subset of B if A ⊆ B and A ≠ B. Two sets are equal if they have exactly the same elements: order and repetition do not matter, so {1, 2, 2, 3} = {3, 2, 1}.

Counting subsets. Each element is either in a subset or not, so a set with n elements has 2ⁿ subsets, of which 2ⁿ − 1 are proper (and 2ⁿ − 2 are non-empty and proper). The number of subsets with exactly k elements is $(k n)$ . The number of subsets of an n-element set that contain a given set of m elements is 2ⁿ⁻ᵐ. Subsets with an odd number of elements and with an even number each total 2ⁿ⁻¹.

Power set. P(A) is the set of all subsets of A; n(P(A)) = 2ⁿ. For A = {a, b} we get P(A) = {∅, {a}, {b}, {a, b}}.

Intervals. For real numbers a < b: open (a, b) = {x : a < x < b}, closed [a, b], half-open [a, b) and (a, b]. Infinite intervals such as (−∞, a] and (a, ∞) are written with ∞ always open. On the number line a filled dot shows an included end point and a hollow dot an excluded one. Set operations on intervals are done by drawing both on one number line and watching the end points carefully: an end point is in the union if it is in either interval, and in the intersection only if it is in both.

Universal set and complement. All sets under discussion are subsets of a universal set U. The complement $A^{'}$ = U − A.

Subsets, Power Set and Intervals: formulas to remember

Number of subsets: $$2^n$ subsets | $2^n - 1$ proper subsets$; For a set with n elements.
Subsets of size k: $\binom{n}{k}$; Sum over k is $2^{n}$ .
Subsets containing m given elements: $2^{n-m}$; The rest are free.
Intervals: $(a,b)$, $[a,b]$, $[a,b)$, $(a,b]$, $(-\infty, a]$; Infinity is never included.

Subsets, Power Set and Intervals in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Choosing clubs

Suppose a school offers 5 clubs and a student may join any collection of them, including none.

Find the number of possible collections, those with at least one club, those with exactly 2 clubs and those with at most 2 clubs.

Try it first, then show the solutionSolution

Each club is in or out: $2^{5} = 32$ collections.
At least one club: $32 - 1 = 31$ . Exactly 2 clubs: $(2 5) = 10$ .
At most 2: $1 + 5 + 10 = 16$ .

Answer: 32; 31; 10; 16.

Memory hook: Every element has two choices: in or out.

Tricks to remember Subsets, Power Set and Intervals

Short memory aids. Say each one aloud once, then use it on a question the same day.

In or out
Each element has two choices, so multiply 2 for each element.
Use it when: Counting subsets.
Empty set counts
The empty set is a subset of every set, and it is a proper subset of every non-empty set.
Use it when: Proper-subset counts.

Subsets, Power Set and Intervals: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Proper subsets · 1 mark · MCQBoard level

The number of proper subsets of A = {1, 2, 3, 4} is:
(a) 16
(b) 15
(c) 14
(d) 8

Try it first, then show the solutionSolution

A has $2^{4} = 16$ subsets; excluding A itself gives 15 proper subsets.

Answer: (b) 15

Example 2: Power set · 3 marks · Short answerBoard level

Write the power set of A = {a, b, c} and state the number of its elements.

Try it first, then show the solutionSolution

The subsets of size 0: $\emptyset$ . Size 1: {a}, {b}, {c}. (1 mark)
Size 2: {a, b}, {a, c}, {b, c}. Size 3: {a, b, c}. (1 mark)
$n (P (A)) = 2^{3} = 8$ . (1 mark)

Answer: P(A) has 8 elements: ∅, {a}, {b}, {c}, {a,b}, {a,c}, {b,c}, {a,b,c}.

Example 3: Counting subsets · 4 marks · Short answerHarder board level

Let A = {1, 2, 3, 4, 5, 6}. Find the number of subsets of A (i) that contain 1 but not 2, (ii) that contain both 1 and 2, (iii) with exactly 3 elements, (iv) with an odd number of elements.

Try it first, then show the solutionSolution

(i) 1 is in, 2 is out; the other 4 elements are free: $2^{4} = 16$ . (1 mark)
(ii) Both in: $2^{4} = 16$ . (1 mark)
(iii) $(3 6) = 20$ . (1 mark)
(iv) Half of all subsets have odd size: $2^{5} = 32$ . (1 mark)

Answer: (i) 16; (ii) 16; (iii) 20; (iv) 32.

Example 4: Operations on intervals · 4 marks · Short answerHarder board level

If A = [−2, 5) and B = (1, 7], find A ∩ B, A ∪ B, A − B and B − A.

Try it first, then show the solutionSolution

A ∩ B: points in both, $(1, 5)$ . (1 mark)
A ∪ B: $[- 2, 7]$ . (1 mark)
A − B: points of A not in B, $[- 2, 1]$ (1 is in A and not in B). (1 mark)
B − A: points of B not in A, $[5, 7]$ (5 is not in A). (1 mark)

Answer: (1, 5); [−2, 7]; [−2, 1]; [5, 7].

Example 5: Sets between two sets · 5 marks · Long answerHOTS / competency

A and B are sets with A ⊆ B, n(A) = 3 and n(B) = 5. Find (i) the number of sets C with A ⊆ C ⊆ B, (ii) the number of subsets of B that are not subsets of A, (iii) the number of subsets of B that contain exactly one element of A.

Try it first, then show the solutionSolution

(i) C must contain A and may contain any of the other $5 - 3 = 2$ elements: $2^{2} = 4$ . (1.5 marks)
(ii) All subsets of B are $2^{5} = 32$ ; those that are subsets of A number $2^{3} = 8$ ; difference $32 - 8 = 24$ . (1.5 marks)
(iii) Choose the one element of A in $(1 3) = 3$ ways; the 2 other elements of B are free ( $2^{2} = 4$ ): $3 \times 4 = 12$ . (2 marks)

Answer: (i) 4; (ii) 24; (iii) 12.

Where students lose marks in Subsets, Power Set and Intervals

The slips examiners see most often, and what to write instead.

Mistake: Counting the empty set wrongly
Giving 2ⁿ − 1 for all subsets.
Do this: All subsets: 2ⁿ; proper subsets: 2ⁿ − 1.
Usually costs: 1 to 2 marks.
Mistake: Closed/open end-point errors
Writing [1, 5] for the intersection of [−2, 5) and (1, 7].
Do this: Check each end point in both intervals.
Usually costs: 1 to 2 marks.
Mistake: Treating {a, a} as two elements
Counting repeats.
Do this: A set lists each element once.
Usually costs: 1 mark.

Exam tips

Say "each element is in or out" to justify 2ⁿ.
Draw intervals on one number line.
Mark end points as included or excluded in your working.
List a small power set fully before generalising.

Practice questions on Subsets, Power Set and Intervals

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
If A = {x : x is a prime number less than 10}, the number of subsets of A is:
(a) 4
(b) 8
(c) 16
(d) 32
Hint
List A first: {2, 3, 5, 7}.
Answer
(c) 16
Q2 · 2 marks · Short answerBoard level
Are {1, 2, 2, 3} and {3, 2, 1} equal? Give a reason.
Hint
Repetition and order do not matter.
Answer
Yes. They have exactly the same elements, 1, 2 and 3.
Q3 · 3 marks · Short answerHarder board level
How many subsets of {1, 2, ..., 8} have exactly 3 elements, and how many have an odd number of elements?
Hint
Use $(3 8)$ and the half rule.
Answer
$(3 8) = 56$ . Odd-size subsets: $2^{7} = 128$ .
Q4 · 3 marks · Short answerHarder board level
Write the interval notation for ${x : x \in R, - 3 < x \leq 4}$ and for its complement in ℝ.
Hint
Mind the end points.
Answer
$(- 3, 4]$ ; the complement is $(- \infty, - 3] \cup (4, \infty)$ .
Q5 · 3 marks · Short answerHarder board level
Let A = {1} and let P(A) be its power set. Find n(P(P(A))), and list P(A).
Hint
Build up one step at a time.
Answer
P(A) = {∅, {1}} has 2 elements, so P(P(A)) has $2^{2} = 4$ elements.
Q6 · 4 marks · Case-basedHOTS / competency
Case study: Suppose a coach picks teams from 5 players A, B, C, D and E, and every team must have at least 1 player.
(i) How many teams are possible? (1 mark)
(ii) How many teams include player A? (1 mark)
(iii) How many teams of exactly 3 players? (1 mark)
(iv) How many teams include A but not B? (1 mark)
Hint
Teams are non-empty subsets.
Answer
(i) $2^{5} - 1 = 31$ . (ii) A in: $2^{4} = 16$ subsets, all non-empty: 16. (iii) $(3 5) = 10$ . (iv) A in, B out, three free: $2^{3} = 8$ (all non-empty because A is in).

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.Number of subsets of an n-element set?Show answer
$2^{n}$ .
2.Is ∅ a subset of every set?Show answer
Yes.
3.Number of subsets with k elements?Show answer
$(k n)$ .
4.Is ∞ ever included in an interval?Show answer
No.

Subsets, Power Set and Intervals: frequently asked questions

What is the difference between ⊂ and ∈?

a ∈ A says a is an element of A; A ⊂ B says every element of A is in B. An element and a subset are different things.

Is ∅ a proper subset of ∅? Is it a proper subset of {1}?

Not of ∅: ∅ ⊆ ∅ but they are equal. It is a proper subset of every non-empty set, so ∅ is a proper subset of {1}.

What is the cardinality of P(P(A))?

2^{2^n} for n(A) = n.