Subsets

What is a Subset?

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

Symbol: $\subseteq$

Definition

$A \subseteq B \iff \text{for all } x, \text{ if } x \in A \text{ then } x \in B$

“A is a subset of B” means every element of A is also in B.

Example

$A = \{1, 2\}$ $B = \{1, 2, 3, 4\}$

Is A a subset of B?

Check each element of A:

• Is 1 in B? Yes ✓
• Is 2 in B? Yes ✓

All elements of A are in B, so:

$A \subseteq B$

More Examples

Two Surprising Facts

1. Every set is a subset of itself

$A \subseteq A$

Why? Every element of A is in A. Trivially true.

2. The empty set is a subset of every set

$\emptyset \subseteq A \text{ for any } A$

Why? There are no elements in $\emptyset$ that could fail to be in A.

This is called a vacuously true statement — true because there’s nothing to contradict it.

Proper Subsets

Sometimes we want to say A is “strictly inside” B — not equal to B.

Proper subset: A is inside B, but A ≠ B.

Symbol: $\subset$ (no line underneath)

Examples:

The difference:

• $A \subseteq B$ — A is inside B, or A equals B
• $A \subset B$ — A is inside B, and A is smaller than B

Subset vs Element

Don’t confuse $\in$ and $\subseteq$!

Element ($\in$): A single thing is in a set

$2 \in \{1, 2, 3\}$

Subset ($\subseteq$): A whole set is inside another set

$\{2\} \subseteq \{1, 2, 3\}$

Properties

Reflexive:

$A \subseteq A$

Every set is a subset of itself.

Transitive:

$\text{If } A \subseteq B \text{ and } B \subseteq C, \text{ then } A \subseteq C$

Subsets chain together.

Antisymmetric:

$\text{If } A \subseteq B \text{ and } B \subseteq A, \text{ then } A = B$

If they’re subsets of each other, they’re equal.

Proving Set Equality

To prove two sets are equal, prove both directions:

1. Show $A \subseteq B$ (every element of A is in B)
2. Show $B \subseteq A$ (every element of B is in A)

If both are true, then $A = B$.

Summary:

• $\subseteq$ means “is inside or equal to”
• $\subset$ means “is strictly inside”
• $\emptyset$ is the empty set — a set with no elements: $\emptyset = \{\}$
• $\emptyset$ is a subset of everything
• Every set is a subset of itself