Problem 97 (difficulty: 4/10)

Let \(\displaystyle f:A\to B\). For any set \(\displaystyle X\subset A\) let \(\displaystyle f(X)=\{f(x): ~ x\in X\}\) (the image of the set \(\displaystyle X\)), and for any set \(\displaystyle Y\subset B\) let \(\displaystyle f^{-1}(Y)=\{x\in A: ~ f(x)\in Y\}\) (the preimage of the set \(\displaystyle Y\)). Is it true that

(a) \(\displaystyle \forall X,Y\in \mathcal P(A) ~ f(X)\cup f(Y) = f(X\cup Y)\) ?

(b) \(\displaystyle \forall X,Y\in \mathcal P(B) ~ f^{-1}(X)\cup f^{-1}(Y) = f^{-1}(X\cup Y)\) ?

Give me another random problem!