I try to express the following statements in first order logic:
- X is a subset of Y.
- A set can be uniquely characterised by its elements.
- The power set p(X) contains all subsets of X.
- A set X is the union of all its subsets containing just one element.
Thus far I managed:
- $\forall x \in X \Rightarrow x \in Y$
- $X=Y \Leftrightarrow (\forall x \in X \Rightarrow x \in Y \land \forall y \in Y \Rightarrow y \in X)$
- $ x \in p(X) \Leftrightarrow (\forall y \in x \Rightarrow y \in X) $
However now I don't know to express the 4th problem via FOL.
I tried: $ X = \cup x.(\forall z \in X \exists t. z \in t \land t = x \land z \not \in \forall y. y \neq t)$
If possible I would also like to convert 3. in an expressing $p(X)$ with $=$ rather than $\Leftrightarrow$
Any form of constructive comments are welcome. Thanks in advance.
EDIT: Regarding problem 4.: My main problem is that I am not really sure, how to express or write down the solution - I am not too sure, what is allowed, what not. Thus I take any idea that seems remotely right or constructive.
$ X = \underset{|x| = 1, x \subset X} \cup \Leftrightarrow \forall y \in X \exists x . (y \in x \land \lnot \exists z .(z \lnot = x \land y \in z) ) $
On the LHS of $\Leftrightarrow $ I simply tried to express the property given in the problem above and on its RHS I tried to express this property in terms of FOL.