Subset Formula

A set A is a subset of set B — written A ⊆ B — if every element of A is also in B.

The Formula

A⊆B⇔∀x (x∈A⇒x∈B)

When to use: Every single thing in A can also be found inside B. Think of A as fitting entirely within B, like a small circle inside a big one.

Quick Example

{1,2}⊆{1,2,3} Also, {1,2,3}⊆{1,2,3} (every set is a subset of itself).

Notation

A⊆B means A is a subset of B

What This Formula Means

Set A is a subset of set B if every element of A is also an element of B, written A⊆B.

Every single thing in A can also be found inside B. Think of A as fitting entirely within B, like a small circle inside a big one.

Formal View

A⊆B⇔∀x (x∈A⇒x∈B)

Worked Examples

Example 1

easy
Let A={1,2,3,4,5} and B={2,4}. Determine whether B⊆A.

Answer

B⊆A

First step

1
Recall the definition: B⊆A means every element of B is also an element of A. We check each element of B individually.

Full solution

  1. 2
    Check 2∈B: is 2∈A={1,2,3,4,5}? Yes. Check 4∈B: is 4∈A? Yes.
  2. 3
    Since every element of B belongs to A, we conclude B⊆A. Note also B≠A since A has elements (1, 3, 5) not in B, so B is a proper subset: B⊊A.
B is a subset of A if every element of B belongs to A. We verify this by checking membership one element at a time.

Example 2

medium
List all subsets of S={a,b,c}.

Example 3

medium
List all subsets of {1,2} that contain 1.

Common Mistakes

  • Declaring A⊆B after finding just one common element — every element of A must be in B.
  • Forgetting ∅⊆A for every set — the empty set is a subset of everything, vacuously.
  • Confusing A⊆B with A∈B — subset relates two sets; membership relates an object to a set.

Why This Formula Matters

Subset is how mathematicians prove two sets are equal (show each is a subset of the other) and how they define power sets and partial orders. A student who confuses ∈ with ⊆, or forgets that the empty set is a subset of everything, will stumble on proofs and counting subsets. Recognizing it by "Is every single member of the first set also a member of the second?" — rather than by familiar numbers — is what lets a student tell it apart from element (∈) and proper subset (⊊) and intersection in a mixed problem set.

Frequently Asked Questions

What is the Subset formula?

Set A is a subset of set B if every element of A is also an element of B, written A⊆B.

How do you use the Subset formula?

Every single thing in A can also be found inside B. Think of A as fitting entirely within B, like a small circle inside a big one.

What do the symbols mean in the Subset formula?

A⊆B means A is a subset of B

Why is the Subset formula important in Math?

Subset is how mathematicians prove two sets are equal (show each is a subset of the other) and how they define power sets and partial orders. A student who confuses ∈ with ⊆, or forgets that the empty set is a subset of everything, will stumble on proofs and counting subsets. Recognizing it by "Is every single member of the first set also a member of the second?" — rather than by familiar numbers — is what lets a student tell it apart from element (∈) and proper subset (⊊) and intersection in a mixed problem set.

What do students get wrong about Subset?

The procedure for subset is the easy part; the trap is declaring A⊆B after finding just one common element. Asking "Is every single member of the first set also a member of the second?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Subset formula?

Before studying the Subset formula, you should understand: set, element.

← Back to Subset See All Examples Practice Problems

Related Concepts

SetElement