Question

Let $f:X\to Y$ and $A,B$ are non-void subsets of $Y$, then (where the symbols have their usual interpretation)

• A. $f^{-1}\left(A\right)-f^{-1}\left(B\right)\supset f^{-1}(A-B)$ but the opposite does not hold.
• B. $f^{-1}\left(A\right)-f^{-1}\left(B\right)\subset f^{-1}(A-B)$ but the opposite does not hold.
• C. $f^{-1}(A-B)=f^{-1}\left(A\right)-f^{-1}\left(B\right)$
• D. $f^{-1}(A-B)=f^{-1}\left(A\right)\cup f^{-1}\left(B\right)$

Answer 1

The correct option is C $f^{-1}(A-B)=f^{-1}\left(A\right)-f^{-1}\left(B\right)$

Let $x\in f^{-1}\left(A\right)-f^{-1}\left(B\right)$
$\Rightarrow x\in f^{-1}\left(A\right)$ and $x\notin f^{-1}\left(B\right)$
$\Rightarrow f\left(x\right)\in A$ and $f\left(x\right)\notin B$
$\Rightarrow f\left(x\right)\in (A-B)$
$\Rightarrow x\in f^{-1}(A-B)$
$\Rightarrow f^{-1}\left(A\right)-f^{-1}\left(B\right)\subset f^{-1}(A-B)...\left(1\right)$

Next, we check if $f^{-1}(A-B)\subset f^{-1}\left(A\right)-f^{-1}\left(B\right)$
Let $x\in f^{-1}(A-B)$
$\Rightarrow f\left(x\right)\in (A-B)$
$\Rightarrow f\left(x\right)\in A$ and $f\left(x\right)\notin B$
$\Rightarrow x\in f^{-1}\left(A\right)$ and $x\notin f^{-1}\left(B\right)$
$\Rightarrow x\in f^{-1}\left(A\right)-f^{-1}\left(B\right)$
$\Rightarrow f^{-1}(A-B)\subset f^{-1}\left(A\right)-f^{-1}\left(B\right)...\left(2\right)$

From $\left(1\right)$ and $\left(2\right)$
$f^{-1}(A-B)=f^{-1}\left(A\right)-f^{-1}\left(B\right)$