# Subset Relation — Set Theory

*This is a continuation of my previous post about Set Theory. If you have not read that yet, start **there**.*

When one set contains all of the elements of another set (and may be more), second set is considered to be a subset of the first set. This definition allows, two equivalent sets to be subsets of each other.

`A = {a}`

and`B = {a, b}`

then`A ⊆ B`

`A = {a, b}`

and`B = {a, b}`

then`A ⊆ B`

and`B ⊆ A`

`odd_numbers ⊆ ℤ`

`Formal Notation`

`A ⊆ B ⇔ ∀x (x ∈ A ⇒ x ∈ B)`

This reads as `A`

is a subset of `B`

if and only if `B`

contains every element that is an element of `A`

.

### Proper Subsets

When one set contains all of the elements of another set and some more, then the second set is considered to be a proper subset of the first set.

`A = {a}`

and`B = {a, b}`

then`A ⊂ B`

`Partial Ordering`

Following three rules states that subset is a partial ordering.

#### Reflexivity

`A ⊆ A`

Given the formal notation for subset is`A ⊆ B ⇔ ∀x (x ∈ A ⇒ x ∈ B)`

it is clear that every element present of set `A`

is present in itself. Hence, it can be concluded that `A`

is a subset of itself.

#### Antisymmetry

`A ⊆ B ∧ B ⊆ A ⇒ A = B`

Again, it is clear from the subset definition that if all elements of set `A`

occurs in set `B`

and vice versa, they must be equivalent.

`Transitivity`

`A ⊆ B ∧ B ⊆ C ⇒ A ⊆ C`

Lastly, it is trivial to prove that when set `B`

contains all elements of set `A`

and set `C`

contains all elements of set `B`

, set `C`

in fact contains all the elements of set `A`

.

### Subset Characterisation

`A ⊆ B ⇔ A ∩ B = A`

It is trivial to see that `A`

can be a subset of another set `B`

, if and only if their intersection results in set `A`

.

`A ⊆ B ⇔ A ∪ B = B`

Equally, it is obvious that the set `A`

will only be a subset of `B`

, if and only if their union results in set `B`

.

In both case, the set`A`

is wholly contained within the set`B`

in a Venn diagram.

*Originally published at **medium.com** on September 14, 2017.*