Learning Objectives
- explain what a subset is;
- use ⊆ and ⊂ correctly;
- distinguish subsets from proper subsets;
- list every subset of a small set;
- calculate the number of possible subsets.
Introduction
This lesson develops Subsets from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.
Key Notes
Read the definitions and key facts below carefully. Each rule is most useful when you can explain why it fits the situation, not simply recall it.
🌍 Why This Matters
Subsets describe how sets fit inside one another. This idea is essential when reading Venn diagrams and working with probability.
🤔 Think About This...
Let A be the students in a school and B be the Form 2 students in that school.
B is a subset of A.
What Is a Subset?
B is a subset of A when every element of B is also an element of A.
Let A = {1, 2, 3, 4} and B = {2, 4}.
Both 2 and 4 belong to A, so B ⊆ A.
A subset sits completely inside its larger set
B ⊆ A because every football player shown in B is also a learner who plays sport in A.
Proper Subsets
B is a proper subset of A when every element of B belongs to A, but B is not equal to A.
| Statement | True? | Reason |
|---|---|---|
| {1, 2} ⊆ {1, 2, 3} | Yes | All elements occur in the larger set |
| {1, 2, 3} ⊆ {1, 2, 3} | Yes | Every set is a subset of itself |
| {1, 2, 3} ⊂ {1, 2, 3} | No | A proper subset must be smaller |
The Empty Set Is a Subset
The empty set is a subset of every set because it contains no element that can violate the subset condition.
Listing All Subsets
List all subsets of A = {a, b}.
Include the empty set: ∅.
Include each one-element subset: {a}, {b}.
Include the whole set: {a, b}.
The subsets are ∅, {a}, {b}, {a, b}.
How Many Subsets?
Here n is the number of elements in the original set.
A set contains 4 elements.
Number of subsets = 2⁴ = 16.
⚠️ Common Mistakes
✅ Quick Check
💡 Exam Tip
Remember that every set is a subset of itself, and the empty set is a subset of every set. A proper subset must be smaller than the original set.
🧠 Let’s Think Together
Before calculating, identify the information given, the result required and the mathematical relationship that connects them. Predict whether the answer should be larger, smaller or unchanged, then use that prediction to check the result.
Practice Questions
Foundation- Let A = {1, 2, 3, 4}. State whether:
- {1, 3} ⊆ A;
- {2, 5} ⊆ A;
- A ⊆ A.
- List all subsets of {x, y}.
- How many subsets does a 5-element set have?
- List all proper subsets of {1, 2, 3}.
- Explain the difference between 3 ∈ A and {3} ⊆ A.
- A set has 64 subsets. How many elements does it have?
✅ Check Your Work
Show practice answers
Foundation
- True
- False
- True
- ∅, {x}, {y}, {x, y}
- 2⁵ = 32
Confidence
- ∅, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}
- 3 ∈ A means 3 is an element. {3} ⊆ A means the one-element set {3} is a subset.
Challenge
- 6 elements, because 2⁶ = 64.
Exam Focus
Show the mathematical relationship you are using, keep notation consistent and give a clear final answer. Use a quick estimate, diagram or inverse operation to check that the result is reasonable.
Summary
- B ⊆ A means every element of B belongs to A.
- A proper subset is not equal to the original set.
- ∅ and the whole set are always subsets.
- An n-element set has 2ⁿ subsets.