1
$\begingroup$

I'm trying to read "Set Theory" by Thomas Jech and i'm confused by statement in page 14 (1.14) It states:

The separation Axioms follow from the Replacement Schema.
(1) Given $\varphi$, let $F = \{(x,x):\varphi(x)\}$.
(2) Then $\{x\in X:\varphi(x)\}=F(X)$, for every $X$.

What I don't understand is how (2) follows form (1), specifically:
if we have set $S = \{e_1, e_2, e_3 \dots \}$ and if all elements are not set themselves then
form (1) and $F\cap S= F(S)$ assuming all elements of $S$ satisfy $\varphi$ we get $\{\{\{e_1\}\},\{\{e_2\}\},\{\{e_3\}\} \dots\}$, set of singleton sets. This is from definition of ordered pair and singleton set:
$(a,b) = \{\{a\},\{a,b\}\}$ then
$(a,a) = \{\{a\},\{a,a\}\} = \{\{a\},\{a\}\} = \{\{a\}\}$
but from (2) we clearly get set of original elements, or set $S$.
What do i miss here?

Edit1:
Is nontation $F= \{(x,y):\varphi(x,y,p)\}$ shorthand or convention for something? And if it is for what exactly?

$\endgroup$
2
$\begingroup$

I think there's some confusion over the notation "$F(X)$": by definition, $F(X)=\{y: \exists x\in X((x, y)\in F)\}$. It should then be clear that $F(X)=\{x: \varphi(x)\}$: we have $$x\in F(X)\iff \exists y\in X((x, y)\in F)\iff \exists y\in X(y=x\wedge \varphi(x))\iff x\in X\wedge \varphi(x).$$

$\endgroup$
  • $\begingroup$ In other words, $F$ is the restriction of the identity to the class $\{\,x:\phi(x)\,\}$. $\endgroup$ – Hagen von Eitzen Jul 21 '15 at 6:19
  • $\begingroup$ @Noah thank you for quick answer. You are correct there was confusion over notation of "F(X)" $\endgroup$ – Kuna Prime Jul 21 '15 at 7:11

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.