# Subset Relation

Introduction :

Let A and B be two sets. If every element of A is also present in an element of B, then A is called a subset of B or B is a super set of A.

We can write this detail by the symbol A ⊆ B or B ⊇ A. Here the symbol ⊆ denotes subset and ⊇ denotes a super set of or contains.

## Proper and Improper Subset

Let A and B are two sets. If A is a subset of B and there is an element u ∈ B and u ∉ A, then the set A is called a proper subset of B and it is denoted by A ⊂ B. In this situation, we also write B ⊃ A. For example, let X = {6, 7, 8, 9} Y = {6, 7, 8, 9, 5}. We observe that X is a subset relation of Y and 5 ∈ Y but 5 ∉ X. Therefore, X is a proper subset relation of Y; that is, X ⊂ Y .

Although A is a subset relation of A, it is not a proper subset of A. Hence it is called an improper subset of A.

## Example Problems

Subset relation problem 1:

Find the subset A = {1, −1, 5, −2, 9} and B = {1, 5, −1, −2, 9, −3}.

We observe that 1, 5, −1, −2, 9 ∈ A and 1, 5, −1, −2, 9 ∈ B.

That is, each and every element of the set A is also presented in an element of the set B. So A is a subset relation of B, that is, A ⊆ B.

Subset relation problem 1:

Give any example problem for subset.

Solution:

The symbol ⊂ or ⊃, is used to denote a proper subset relationship:

A = {s, t, u} B = {s, t, u, v, w}

A ⊂ B

Subset relation problem 2:

Give any example problem for not a subset.

Solution:

The symbol ⊄ is used to denote "not a subset":

If  A = {m, n, o} B = {m, n, p, q}

A  ⊄ B