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.