## 4.6.Subset

Let A and B be any two sets defined by

```                       A    = { 3, 4 }
B    = { 3, 4, 5 }
```
Every member of A is also a member of B.
We say that
```                    A is included in B    or,
A is a subset of B.
```

Definition 4.2 (Subset)

A set S is included in another set T if and only if every member of S is a member of T. In this case we also say that S is a subset of T

Venn diagram

To express formally the fact that S is a subset of T, we write S T

Definition 4.3 (Formal counter part)

For any two sets S and T, S T: { for all x S: x S x T }.

Definition 4.4 (Strict subset)

For any sets S and T, S T if and only if S T and there exists x such that x T and x S (which implies S not equal T).
If S T then S is a strict or proper subset of T.

Venn diagram

Definition 4.5 (Empty set)

The empty set is a set with no members, it is denoted by Ø. Thus, formally, Ø is defined by the condition that for any object x, x Ø.

Principle of extensionality

Let A and B any two sets. If A B and B A then A = B.