axiomatic sets 11 (making a subset)

Making a subset $y$  of a set $x$ , we will use a formula $P(z)$ in the preceding post .

It is a statement of properties which all elements $z$ of the subset $y$ has.
\[ \forall x\exists y \forall z[z\in y \leftrightarrow z\in x\wedge P(z)] \]

That is to say,
\[ z\in y=\left\{z\in x : P(z)  \right\} \]
It is called axiom schema of separation, abstruction or subset.

It is not an axiom because there can be many statements $P(z)$ .

Although $P(z)$  must satisfy the promise, as we can not write all $P(z)$  of subsets,
Axiom schema has been used.

As a simple example, let $P(z)$  be $z\in w$ . Then,
\[  y=\left\{ z\in x : z\in w \right\} .  \]
We call $y$ the intersection of $x$  and $w$ , and
we write $y=x\cap w$ .
Therefore, $x\cap w=w\cap x$ .

You　must note that $\left\{z\in x : P(z)  \right\}$ is different to $\left\{z : P(z)  \right\}$ .

Axiom schema of separation is very important for avoiding the paradoxes.

（ This is the 100th post. ）

axiomatic sets 10 (a formula)

In naive set theory, we wrote the set of real numbers in the interval $[0,1]$
like  $\left\{x\in\mathcal{R} | x\in [0,1]   \right\}$ .

The formula $x\in [0,1]$  is the property or the condition which the elements $x$ have.

A set is a collection of some objects.
(there are also some cases in which a collection is not a set and becomes a class. )

We have to state various kinds of properties which elements of a set have for defining the set.
It is a formula $P(x)$ .

IF $P(x)$ is a formula (a statement of properties of $x$ ),
then logical notations below can be only accepted in $P(x)$ ;

belongs to : $\in$
or : $\vee$
and : $\wedge$
not : $\neg$
If then : $\rightarrow$
for all : $\forall$
for any : $\exists$

Occasionally,  $E!(A)$ means " there exists only one A" .

Of course, new notations which have been derived by above are possible.
For example, $\subset$ , $\cup$ ,and so on.

We can also write $\neg(x\in a)$  $x\notin a$ .

This is not an axiom. It is a promise.