1. ## Axiom Question

Any ideas?

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

2. Two things come immediately to mind:

1) Is B not A?
2) Define "exists". "Contains at least one element"?

3. 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.

4. 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 $\displaystyle B$)