# Concept of Subset

The concept of a subset is defined as a set $$A$$ which is said to be the subset of a set $$B$$ if every element of set $$A$$ is also an element of set $$B$$. This relationship is usually denoted by $$A \subset B$$, and mathematically this relationship is written as if $$x \in A$$ implies $$x \in B$$. The concept of a subset is also written in the from of $$A \subseteq B$$

If $$A \subset B$$, in this case we say that $$A$$ is contained in $$B$$ or $$B$$ contains $$A$$. If $$A$$ is a subset of $$B$$ and $$B$$ has at least one element which is not in $$A$$, then $$A$$ is called the **proper subset** of $$B$$. The proper subset more briefly is also defined as $$A$$ is a proper subset of $$B$$ if $$A \subset B$$ and $$A \ne B$$.

**For example**,

$$A =$$ {2, 3, 4, 5, 6, 7, 8} is proper subset of $$B =$$ {1, 2, 3, 4, 5, 6, 7, 8}.

If $$A$$ is a subset of $$B$$, then $$B$$ is called the super set of $$A$$. The symbol $$ \subset $$ is called the inclusion symbol. If $$A$$ is not a subset of $$B$$, we write $$A \not\subset B$$. By the definition of a subset, it is clear that the empty set and the set $$A$$ itself are always subsets of $$A$$. These two subsets are called the **improper subsets** of $$A$$.

If we have to show that $$A$$ is a subset of $$B$$, then we take any arbitrary point of $$A$$ and show that this point also lies in $$B$$. This will give us information that every point of $$A$$ is also a point of $$B$$. Consequently we conclude that $$A$$ is a subset of $$B$$. This concept can be explained as mathematically if $$x \in A$$ implies $$x \in B$$.

The other way to prove that $$A$$ is a subset of $$B$$ is that we show that $$B$$ is a super set of $$A$$. For this purpose, we consider an arbitrary point which is not in $$B$$ and show that this point is also not in $$A$$. In this way we conclude that a point which is not in $$B$$ is also not in $$A$$, which shows that $$B$$ is a super set of $$A$$.