Can someone re-word the following question in to more basic terms:
'Let $X$ be a finite set. A powerset $P(X)$ of $X$ is the set of all subsets of $X$. Define $\lambda$ as follows $\forall g \in Sym(X)$ and $\forall \{x_1, ..., x_n\} \in P(X)$:
$$\lambda(g)\{x_1,...,x_n\} = \{gx_1,...,gx_n\}$$ Prove that $\lambda$ is an action of $Sym(X)$ on $P(X)$.'
Do I simply have to show:
- $\lambda(e_{Sym(X)})x = x$ for all $x \in X$.
2.$\lambda(fg)x = \lambda(f)(\lambda(g)x)$ for all $f,g, \in Sym(X)$ and for all $x \in X$.
Or is there more to the question?
Any help would be greatly appreciated in helping me understand the question a little bit more.
I will answer just this part of the question
If you have a way to permute the elements of a finite set $X$ then there is a natural way to permute subsets of $X$ - just permute what they contain.
You are asked to fill in the formal argument that shows that natural definition really works.
I haven't read your formal argument but I think there's a problem. The assertions should be about "all $A \in P(X)$", not "all $x \in X$".