Vrftsjtryk

In set theory

Axiom Schema of Separation: If $P$ is a property (with parameter$p$),
then for any $X$ and $p$ there exists a set $Y = lbrace u ∈ X : P(u, p) rbrace$ that contains all those $u ∈ X$ that have property $P$.

But some of the books are written as follows:

Axiom Schema of Separation: If $P(u)$ is a property
then for any $X$ and $P(u)$ there exists a set $Y = lbrace u ∈ X : P(u) rbrace$ that contains all those $u ∈ X$ that have property $P(u)$.

But I do't know that what's this $P (u)$ . what is its definition mathematically of $P(u)$ (and $P(u ,p_1,p_2,...,p_n)$?

$P$ is a property: a statement that takes some parameter(s), and is either true or false, depending on those parameters, $P(u)$ is the result of evaluating $P$ with the parameter $u$ (and the same for any other collection of parameters that are valid inputs into $P$).

$p(u)$ can be anything, really, as long as it can be stated with some kind of logic formula.

For example, if $X$ is the set of natural numbers, then $p(u)$ could express the property that $u$ is an even number. But you could also have it express that $u$ is an odd number. Or that it is a prime number. Or that $u$ is a prime number that is the sum of three odd numbers. Or ... like I said, pretty much anything you want.

Note that if you use $u=u$ for the formula $p(u)$, then the subset $Y$ is just the same as $X$ itself, as $u=u$ is true for any $u$. And, if you use the formula $u not = u$, you end up with the empty set.

The way I read this $P$ could be any property. "is an integer divisible by $7$" for example. Or "eats shoots and leaves".

All the schema is saying that given any set, there exists a subset of all elements with the property.

For example. If $P$ = "is divisible by $7$" and $Q$ = "eats shoots and leaves" and $A = $ set of all perfect squares. And $B =$ set of all black and white animals then there is $E = {xin A|P(x)} =$ set of all perfect squares that are divisible by $7={49k^2| kin mathbb Z}$.

There is a $F = {x in A|Q(x)} =$ set of all perfect squares that eat shoots and leaves = $emptyset$.

And there is $G = {xin B|P(x)} = $ set of all black and white animals that are divisible by $7 = emptyset$.

And there is $H = {x in B|Q(x)}=$ set of all black and white animals that eat shoots and leaves = {pandas, ... and I'm sure something else....}

I give you one exemple :

$P(f,x_0):forall varepsilon>0,exists eta>0, forall xin mathcal{D}_fqquad |x-x_0|<eta implies |f(x)-f(x_0)|<varepsilon$

If $f:xto lfloor xrfloor$ and $x_0:=1$ then $P(f,x_0)$ is false, indeed $f$ is not continuous on $x_0=1$

If $f:xto x^2$ and $x_0:=1$ then $P(f,x_0)$ is true

$f,x_0$ are free variables, $varepsilon,eta,x$ are bound variables

Note that we consider $P(u):ule u^2$

$P(1)$ is true

$forall uin mathbb{R}, P(u)$ is wrong, and $u$ is a bound variable and this proposition doesn't depend on $u$