# Concept of Subset

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

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

**For example**,

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

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

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

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