Q.Write down all the subsets of the following sets
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🔒 Start your 14-day free trial to unlock the full solution →Concept understanding — Subset Listing
Subset Listing: A First Look
Let's build this from the ground up — no jargon, just intuition first.
1. The Intuition: What does "subset" mean?
Imagine you have a set — a collection of distinct objects. For example:
Set A = {apple, banana, cherry}
Now, a subset is simply a selection of some (or all, or none) of these objects, taken from the original set.
- You could pick all three → {apple, banana, cherry}
- You could pick just two → {apple, banana}
- You could pick just one → {cherry}
- You could pick none → {} (the empty set)
Each of these is a subset of the original set.
2. The Precise Definition
Definition: A set B is a subset of a set A if every element of B is also an element of A.
We write this as:
B⊆A
If B is not a subset of A, we write:
B⊆A
Key points to remember:
-
Every set is a subset of itself.
Example: {apple, banana} ⊆ {apple, banana}
-
The empty set ∅ (or {}) is a subset of every set.
Why? Because it has no elements, so there's nothing to violate the condition.
-
If B is a subset of A but B=A, we call B a proper subset.
Notation: B⊂A (some books use ⊊)
3. How to "list" all subsets
Subset listing means writing down every possible subset of a given set.
Example: Set S={a,b}
All subsets:
- ∅ (empty set)
- {a}
- {b}
- {a,b} (the set itself)
So the list of all subsets is:
{∅,{a},{b},{a,b}}
How many subsets does a set have?
If a set has n elements, it has exactly 2n subsets.
- n=0 → 20=1 subset (just the empty set)
- n=1 → 21=2 subsets
- n=2 → 22=4 subsets (as above)
- n=3 → 23=8 subsets
Why 2n?
For each element, you have 2 choices: include it or exclude it. Multiply these choices: 2×2×⋯×2 (n times) = 2n.
4. A systematic way to list subsets
For a set with n elements, you can use a binary counting method:
- Label each element with a position (1st, 2nd, 3rd, ...)
- Count from 0 to 2n−1 in binary
- Each binary number tells you which elements to include (1 = include, 0 = exclude)
Example: S={a,b,c} (3 elements)
| Binary | Subset |
|---|---|
| 000 | ∅ |
| 001 | {c} |
| 010 | {b} |
| 011 | {b,c} |
| 100 | {a} |
Why this formula?
Okay, let's break down Subset Listing from the ground up. The core idea is simple: given a set, how do we systematically list all its subsets, and why does the formula 2n work?
1. The Core Question
Imagine you have a set with n elements, like S={a,b,c} (so n=3). A subset is any collection of elements from S, including the empty set {} and the set itself {a,b,c}.
The key formula is:
Total number of subsets of a set with n elements = 2n
Let's see why this is true, not just memorize it.
2. The "Decision" or "Binary Choice" Reasoning
The most intuitive derivation comes from thinking about each element individually.
For each element in the original set, when building a subset, you have exactly two choices:
- Include the element in the subset.
- Exclude the element from the subset.
This is a fundamental, independent decision for every element.
Example with S={a,b,c}
- For element a: Choose IN or OUT. (2 choices)
- For element b: Choose IN or OUT. (2 choices)
- For element c: Choose IN or OUT. (2 choices)
Since these choices are independent (choosing for a doesn't affect the choice for b), the total number of distinct combinations of choices is the product of the number of choices for each element:
2×2×2=23=8
This directly gives the 8 subsets of {a,b,c}:
- {} (all OUT)
- {a} (a IN, b OUT, c OUT)
- {b}
- {c}
- {a,b}
- {a,c}
- {b,c}
- {a,b,c} (all IN)
3. The General Formula (Derivation)
For a set with n elements, you have n independent binary decisions. Therefore:
Total subsets=n times2×2×⋯×2=2n
This is the fundamental reason the formula holds. It's not a coincidence; it's a direct consequence of the counting principle for independent events.
4. Why This Matters for Exams
- Don't just memorize 2n. If a question asks "How many subsets does a set with 5 elements have?", you can instantly say 25=32. But if they ask why, you now have the reasoning. …
Concept: Subset enumeration using the power-set construction.
A subset of a set S is any collection of elements from S, including the empty set ϕ and S itself. For a set with n elements, there are exactly 2n subsets.
(i) {a} has 21=2 subsets: ϕ,{a}
(ii) {a,b} has 22=4 subsets: ϕ,{a},{b},{a,b}
(iii) {1,2,3} has 23=8 subsets: ϕ,{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}
(iv) ϕ has 20=1 subset: ϕ (the empty set is a subset of itself) …
A subset is any collection of elements from a set, including the empty set and the set itself. For a set with n elements, there are exactly 2n subsets.
Why Subsets Work This Way
When we form subsets, we're making a yes-or-no decision for each element: include it or leave it out. This binary choice for every element explains why a set with n elements has 2n subsets. The empty set ϕ is always a subset (we said "no" to everything), and the original set is always a subset of itself (we said "yes" to everything).
The key insight: subsets don't create new elements or change order—they simply select which elements to keep.
Finding All Subsets
(i) Subsets of {a}
This set has 1 element, so we expect 21=2 subsets.
- The empty subset: Choose no elements → ϕ
- The full set: Choose the element a → {a}
The subsets are: ϕ,{a}
(ii) Subsets of {a,b}
This set has 2 elements, so we expect 22=4 subsets.
- Choose neither element: ϕ
- Choose only a: {a}
- Choose only b: {b}
- Choose both elements: {a,b}
The subsets are: ϕ,{a},{b},{a,b}
(iii) Subsets of {1,2,3}
This set has 3 elements, so we expect 23=8 subsets.
We can organize them by size:
| Size | Subsets |
|---|---|
| 0 elements | ϕ |
| 1 element | {1},{2},{3} |
| 2 elements | {1,2},{1,3},{2,3} |
| 3 elements | {1,2,3} |
The subsets are: ϕ,{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}
To avoid missing subsets, list them systematically by size (as shown above) or use binary counting: represent each subset by a binary number where 1 means "include" and 0 means "exclude."
(iv) Subsets of ϕ …
Method: Listing All Subsets (Power Set) of a Finite Set
Method Name: Systematic Listing by Size
Why This Works
Every subset is formed by an independent "include or exclude" decision for each element — with n elements there are 2n such combinations, hence 2n subsets. Listing them by size (empty set, then 1-element, then 2-element, and so on) guarantees none are missed or repeated.
Steps
Step 1: Count the elements
A set with n elements has exactly 2n subsets.
Step 2: Start with the empty set
ϕ is always a subset of every set — list it first.
Step 3: List every 1-element subset
Take each element on its own: {a}, {b}, etc.
Step 4: List every 2-element subset …
🧠 The Core Idea: Subset vs. Element
Before we list mistakes, remember the two key symbols:
- ⊂ (subset): Every element of the first set must be in the second set.
- ∈ (element): The entire thing on the left is a single member of the set on the right.
Mixing these up is the #1 cause of errors.
✗ Common Mistake #1: Confusing ⊂ with ∈
Example from the list:
Statement (v): {a}∈{a,b,c}
Why it’s wrong:
- {a} is a set containing the letter a.
- {a,b,c} contains the elements a, b, and c — not the set {a}.
- So {a} is not an element of {a,b,c}.
✓ Correct thinking:
- {a}⊂{a,b,c} is true (every element of {a} is in the big set).
- {a}∈{a,b,c} is false unless the big set explicitly contains a set as an element, e.g., {a,{a},b}.
How to avoid:
Ask yourself: “Is the left side a single object inside the right side, or is it a collection whose members are inside?”
✗ Common Mistake #2: Forgetting that ⊂ requires all elements
Example from the list:
Statement (iii): {1,2,3}⊂{1,3,5}
Why it’s wrong:
- The left set has 1,2,3.
- The right set has 1,3,5.
- 2 is missing from the right set. So it’s false.
✓ Correct thinking:
- For ⊂ to be true, every element of the first set must appear in the second. One missing element = false.
How to avoid:
Check each element one by one. If even one is missing, the statement is false.
✗ Common Mistake #3: Misreading “not a subset” (⊂)
Example from the list:
Statement (i): {a,b}⊂{b,c,a}
Why it’s wrong:
- The left set has a and b.
- The right set has b,c,a — both a and b are present.
- So {a,b} is a subset. The statement says it is not a subset — that’s false.
✓ Correct thinking:
- {a,b}⊂{b,c,a} is true.
- Therefore {a,b}⊂{b,c,a} is false.
How to avoid:
First check if it is a subset. Then apply the “not” (⊂) to decide true/false.
✗ Common Mistake #4: Overlooking the definition of the set on the right
Example from the list:
Statement (ii): {a,e}⊂{x:x is a vowel in the English alphabet}
Why it’s correct (but often marked wrong by students):
- Vowels: a,e,i,o,u.
- The left set has a and e — both are vowels.
- So it is a subset — true.
Common error: Students sometimes think “vowel” means only a,e,i,o,u but then forget to check if a and e are actually in that list. Or they misread the set-builder notation.
How to avoid:
Write out the actual elements of the set described in words. Then compare.
✗ Common Mistake #5: Not simplifying the set before comparing
Example from the list: …
Showing the 12 most recent of 26 on this concept.
- CBSE 2026Set ANNUAL1 markMCQQ.The number of all non-empty subsets of the set {1,2,3} is:(a) 8(b) 7(c) 3(d) 9
›Reveal solutionSolution
A set of n elements has 2n subsets in total; removing the empty subset gives 2n−1 non-empty subsets.
For the set {1,2,3}, n=3.
Total subsets =2n=23=8 (this includes ∅ and the full set itself).
Non-empty subsets =2n−1=8−1=7.
…
- CBSE 2026Set ANNUAL1 markQ.Define subset.
›Reveal solutionSolution
A⊆B means every element of A belongs to B.
Formally, A is a subset of B if, for every element x, x∈A⇒x∈B.
…
- CBSE 2026Set 1A1 markQ.List all the elements of the set C={x:x is an integer,x2≤4}.
›Reveal solutionSolution
Integers with x2≤4 are −2,−1,0,1,2.
We need integers x with x2≤4, i.e. −2≤x≤2. The integers in this r …
- CBSE 2025Set ANNUAL1 markMCQQ.All possible subsets of set A={2,3,4} are(a) P(A)={2},{3},{4}(b) P(A)={{2},{3},{4},{2,3,4}}(c) P(A)={ϕ,{2},{3},{4},{2,3},{3,4},{4,2},{2,3,4}}(d) None of these
›Reveal solutionSolution
A set with n elements has 2n subsets. For A={2,3,4}, n=3, so there are 8 subsets total, exactly the list in option (c).
The power set P(A) is the collection of ALL subsets of A -- including the empty set ϕ and A itself.
Here n(A)=3, so n(P(A))=23=8.
Listing every subset by size:
- Size 0: ϕ
- Size 1: {2},{3},{4}
- Size 2: {2,3},{3,4},{2,4}
- Size 3: {2,3,4} …
- CBSE 2025Set ANNUAL1 markQ.If set A has 4 elements, then number of subsets is .............
›Reveal solutionSolution
The number of subsets of a set with n elements is 2n.
Set A has 4 elements, so the number of subsets is 24=16 (this counts the empt …
- CBSE 2025Set sz1 markMCQQ.If X={0,−1} then P(A) has :(a) 4 elements(b) 3 elements(c) 2 elements(d) 6 elements
›Reveal solutionSolution
Reading the paper's P(A) as the power set of the given set X, a 2-element set has 22=4 subsets.
Honest note on the printed stem: the question states "If X={0,−1} then P(A) has :" — the set introduced is X, but the power-set notation printed is P(A), not P(X). This is almost certainly a printing slip in the original paper (there is no set A defined anywhere in this question), and the only way the question makes sense is if it means the power set of X, i.e. P(X).
Solving for P(X): X={0,−1} has n=2 elements.
…
- CBSE 2025Set ANNUAL1 markMCQQ.Match the column: Column A entry 'Number of subsets of the set {2,3,5}' — find the matching value from Column B.(a) 2(b) 8(c) 32(d) 1−tan2x2tanx(e) sin2x(f) 10(g) 20(h) 1+tan2x2tanx(i) 4
›Reveal solutionSolution
A set with n elements has exactly 2n subsets; here n=3 gives 23=8.
The set {2,3,5} has n=3 elements. The number of subsets of a set with n elements is 2n (this includes the empty set ∅ and the set itself).
…
- CBSE 2024Set ANNUAL1 markMCQQ.If A={2,4,6}, then number of total subsets of A is(a) 8(b) 16(c) 15(d) None of these
›Reveal solutionSolution
The number of subsets of a set with n elements is 2n.
A={2,4,6} has n(A)=3 elements. The total number of subsets of a set (i.e. the number of elements in its power set) is 2n, because each element independently is either included or excluded from a subset.
…
- CBSE 2024Set ANNUAL1 markMCQQ.The number of subsets of a set having n elements is:(a) 2n(b) n2(c) 2n(d) 2n−1.
›Reveal solutionSolution
A set with n elements has exactly 2n subsets.
For a set A={a1,a2,…,an} with n elements, to build any subset we decide, for each element, whether to include it or not — that is 2 independent choices per element. By the multiplication principle, the total number of subsets is …
- CBSE 2024Set sz1 markMCQQ.Subsets of set {−1,1} are:(a) ϕ,{−1}(b) ϕ only(c) ϕ,{−1},{1},{−1,1}(d) 0
›Reveal solutionSolution
A set with n elements has 2n subsets; the set {−1,1} has 2 elements so it has 22=4 subsets: the empty set, the two singletons, and the set itself.
A subset of a set S is any set formed by taking zero or more elements of S (including none, giving the empty set ϕ, and all of them, giving S itself).
For S={−1,1}, which has n=2 elements, the number of subsets is 2n=22=4.
Listing them out:
- Taking 0 elements: ϕ
- Taking 1 element: {−1}, {1}
- Taking 2 elements: {−1,1} …
- CBSE 2024Set ANNUAL1 markMCQQ.The number of subsets of the set {1,2} is:(a) 1(b) 2(c) 3(d) 4
›Reveal solutionSolution
A set with n elements has 2n subsets, so {1,2} (2 elements) has 22=4 subsets.
Step 1. The number of subsets of a set with n elements is 2n, since each element is independently either included or excluded.
…
- CBSE 2024Set ANNUAL1 markQ.Write true or false: A={1,2,3} is a proper subset of B={1,2,3,4}.
›Reveal solutionSolution
A is a proper subset of B because every element of A lies in B, but B has an extra element A lacks.
Step 1. Check A⊆B: every element of A={1,2,3} — namely 1,2,3 — is present in B={1,2,3,4}. So A⊆B.
Step 2. Check A=B: B contains 4, which is not in A. So A=B.
…
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