axiomatic sets 8 (subsets)

By axiom of extensionality, we defined two same sets.

Its condition is all elements of each sets is same.
If $a=\left\{ x,y  \right\}$  and $b=\left\{ x,y  \right\}$ , then $a=b$ .

If there are different elements in two sets,
or there are not some elements in each sets,
then two sets are not same.

Given two sets $a,b$ , if
\[ \forall x[x\in a\rightarrow x\in b] ,  \]
then the set $a$  is called the subset of the set $b$ .
We will write it $a\subset b$ .

Usually $a\subset b$  accepts the case of $a=b$ .

Hence, in addition if
\[ \exists y[y\notin a\wedge y\in b ] , \]
then $a$  is called the proper subset of $b$ .

The proper subset means the elements of the subset is always in the set ,
and there are some elements of the set not in the subset . Therefore, $a\neq b$ .

Please understand the difference of $\in$  and $\subset$ .

Although both are the binary relationship, $\in$  is the relation of a element and a set,
and $\subset$  is the relation of a set and a set.

Given two sets $a=\left\{ x,y  \right\},b=\left\{ x,y,z  \right\}$ .
As $a\subset b$ ($a$  is the proper subset of $b$ ) , $a\notin b$ .

As $x$ (and $y$) is in both $a$ and $b$ , $x\in a$ and $x\in b$ .
However, $x$ are not the subset of $a$  and $b$ ,
because the elements of $x$ are not in $a$ and $b$ .

If $a=\left\{ x,y  \right\},b=\left\{ x,y,\left\{ x,y  \right\}  \right\}$ , then $a\in b$  and $a\subset b$ .

Please note again the difference of $\left\{ x  \right\}$  and $\left\{ \left\{ x  \right\}  \right\}$ .


(This is not axiom of ZFC. We will use in axiom of power set. )