# What is the difference between a subset and a proper subset?

Set theory and Venn diagram are useful tools to understand these two sets in detail.

## Answer: A subset of a set A can be equal to set A but a proper subset of a set A can never be equal to set A.

A proper subset of a set A is a subset of A that** cannot** be equal to A. In other words, if B is a proper subset of A, then all elements of B are in A but A contains at least one element that is not in B.

**Explanation:**

Subset: If A and B are sets and every element of A is also an element of B, then:

A is a subset of B, denoted by A ⊆ B.

or equivalently, B is a superset of A, denoted by B ⊇ A.

For example, A = {1, 2, 3, 4, 5} and B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.

A = {1, 2, 3} and B = {1, 2, 3}

Here, A is a subset of B, or we can say that B is the superset of A.

Proper Subset: If A is a subset of B, but A is not equal to B (that is, there exists at least one element of B which is not an element of A), then

A is also a proper (or strict) subset of B; this is written as A ⊊ B.

For example A = {1, 2, 3} and B = {1, 2, 3, 4}.

Clearly, A is not equal to B and element {4} belongs to set B but is absent in set A, so we have one element in set B which is not an element of set A. Thus, A can be called a proper subset of B.

### Hence, a proper subset of a set A is a subset of A that** is** not equal to A. In other words, if B is a proper subset of A, then all elements of B are in A but A contains at least one element that is not in B, thus the proper subset may or may not be the same.

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