# Axiom Question

• Sep 7th 2009, 06:53 AM
qwe123
Axiom Question
Any ideas?

Show that the set of all x such that x is in A and x is not in B exists.
• Sep 7th 2009, 07:23 AM
TKHunny
Two things come immediately to mind:

1) Is B not A?
2) Define "exists". "Contains at least one element"?
• Sep 7th 2009, 07:57 AM
ThePerfectHacker
Quote:

Originally Posted by qwe123
Any ideas?

Show that the set of all x such that x is in A and x is not in B exists.

Define the property statement P(x) to mean "x is not in B".
By the Axiom Schema of Comprehension there is a set S such that x is S if and only if x in A and P(x) is true, that is, x is in A and x is not in B.
• Sep 7th 2009, 08:46 AM
Taluivren
Hi,
what are $\displaystyle A,B$? sets?
if they are sets, then the axiom schema of restricted comprehension, namely the axiom $\displaystyle \forall b\forall a \exists z \forall x (x\in z \leftrightarrow (x \in a\,\&\, \phi(x,b)))$ where $\displaystyle \phi(x,b)$ is the formula $\displaystyle x \not \in b$, ensures existence of the set you described (just evaluate $\displaystyle a$ to be $\displaystyle A$ and $\displaystyle b$ to be $\displaystyle B$)