Proper Subset

What Is a Proper Subset?

In set theory, a proper subset is a set whose elements are all contained in another set, but which is not identical to that set.

More formally, a set A is a proper subset of a set B if:

• every element of A is also an element of B;
• B contains at least one element that does not belong to A.

In other words, A is completely contained within B, but B has additional elements that are not in A.

This relationship is known as strict inclusion (or proper inclusion).

A proper subset is denoted by the symbol ⊂.

The notation A⊂B is read as "A is a proper subset of B" or "A is strictly contained in B".

Because B contains at least one element that is not in A, the two sets cannot be equal. Therefore, whenever A is a proper subset of B, it follows that A≠B.

Proper subset vs. subset
A proper subset (A⊂B) and a subset (A⊆B) are closely related concepts, but they are not the same.

If A is a proper subset of B, then every element of A belongs to B and the two sets are different. Therefore, A⊂B always implies A⊆B.

However, the reverse is not always true. If A⊆B, the two sets may still be identical. For example, every set is a subset of itself, but no set is a proper subset of itself.

Example

Consider the following sets:

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

$$ B = \{ 1,3,4,2,6,7 \} $$

Every element of A is also contained in B, so the first condition for proper inclusion is satisfied.

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

The second condition is satisfied as well because B contains elements that are not in A.

Specifically, the elements {2,6,7} belong to B but not to A.

$$ B = \{ 1,3,4,\color{red}2,\color{red}6,\color{red}7 \} $$

Since both conditions are satisfied, A is a proper subset of B.

$$ A \subset B $$

This is a simple example of strict inclusion: all the elements of A are contained in B, but B contains additional elements.