Difference Between Subset and Superset

Subset vs Superset

In mathematics, the concept of set is fundamental. The modern study of set theory was formalized in the late 1800s. Set theory is a fundamental language of mathematics, and repository of the basic principles of modern mathematics. On the other hand, it’s a branch of mathematics in its own rights, which is classified as a branch of mathematical logic in modern mathematics.

A set is a well defined collection of objects. Well-defined means, that there exists a mechanism by which one is able to determine whether a given object belongs to a particular set or not. Objects that belong to a set are called elements or members of the set. Sets are usually denoted by capital letters and lower case letters are used to represent elements.

A set A is said to be a subset of a set B; if and only if, every element of set A is also an element of set B. Such a relation between sets is denoted by A ⊆ B. It can also be read as ‘A is contained in B’. The set A is said to be a proper subset if A ⊆ B and A ≠B, and denoted by A ⊂ B. If there is even one member in A that is not a member of B, then A cannot be a subset of B. Empty set is a subset of any set, and a set itself is a subset of same set.

If A is a subset of B, then A is contained in B. It implies that B contains A, or in other words, B is a superset of A. We write A ⊇ B to denote that B is a superset of A.

For an example, A = {1, 3} is a subset of B = {1, 2, 3}, since all the elements in A contained in B. B is a superset of A, because B contains A. Let A={1, 2, 3} and B={3, 4, 5}. Then A∩B={3} . Therefore, both A and B are supersets of A∩B. The set A∪B, is a superset of both A and B, because A∪B, contains all the elements in A and B.

If A is a superset of B and B is a superset of C, then A is a superset of C. Any set A is a superset of empty set and any set itself a superset of that set.

‘A is a subset of B’ is also read as ‘A is contained in B’, denoted by A ⊆ B.

‘B is a superset of A’ is also read as ‘B is contains in A’, denoted by A ⊇ B.