How are inverse images used to prove set inclusions?

[drmarchjune]

I am studying the very first chapter of analysis, but can't quite get through this problem:

Prove f −1(f(A)) ⊇ A
Prove f(f −1(B)) ⊆ B

 

[Stephen Tashi]

How did you begin the problem?
For $A \subseteq f^{-1}( f(A))$, consider two cases.
case 1:$A = \emptyset$
case 2: $A \neq \emptyset$

 

[issacnewton]

in general , to prove that $$A\subseteq B$$ , we prove

$$\forall x[x\in A\Rightarrow x\in B]$$


and to prove this ,you let x be arbitrary. since we have an implication inside the square bracket, we assume antecedent , and set out to prove consequent . So assume
$$x\in A$$