Define the pre-image of a set $S \subseteq Y$ under $f$ where $f:X \to Y$ by $f^{-1}(S) = \{ x \in X : f(x) \in S \}$
Let $A = \{ 0 , 1\}, B = \{ 0,1,2,3 \}$. Define $f:A \to B$ by $f:x \mapsto x + 1$.
According to this definition, is $f^{-1}( \{ 1, 3 \} ) = \{ 0 \} = f^{-1}(\{ 1 \})$ ?
Or is $f^{-1}( \{ 1, 3 \} ) = \emptyset $ ?