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 $A$.