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Important to note that $B$ and $C$ are subsets of $Y$ where $f: X \to Y$

This appears to be a direct proof $$y \in f^{-1}(C/B)$$ implies that $$y \in f^{-1}(C)$$ and $$y \notin f^{-1}(B)$$ Therefore $$y \in f^{-1}(C)/f^{-1}(B)$$

Of course this must be proved backwards as well:

$$y \in f^{-1}(C)/f^{-1}(B)$$

implies

$$y \in f^{-1}(C), y \notin f^{-1}(B)$$

$$y \in f^{-1}(C/B)$$

The proof may need to be cleaned up but is the logic sound?

Martin Sleziak
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Oberyn
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1 Answers1

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Not quite. $x \in f^{-1}[C\setminus B]$ ( notice the \setminus symbol for set difference, forward slash / is for quotients! ) iff $f(x) \in C \setminus B$ iff $f(x) \in C$ and $f(x) \notin B$ iff $x \in f^{-1}[C]$ and $x \notin f^{-1}[B]$ iff $x \in f^{-1}[B]\setminus f^{-1}[C]$.

So you need to talk about $x$ (as the sets live in the domain) and its image $f(x)$, not $y$, that's confusing. We repeatedly use the definition $f^{-1}[A] = \{x \in X: f(x) \in A\}$, plus the definition of set difference. In the above sequence of iff statements, as an exercise, state next to each equivalence to which definition this is due.

Henno Brandsma
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  • Okay I see what you are saying with x and f(x) so how i do make that flow? Is it okay to show it with iif in my presentation? – Oberyn Sep 12 '19 at 22:42
  • @Oberyn it’s allowed to use iff when you justify it. Your first step e.g. is not reasoned. – Henno Brandsma Sep 13 '19 at 04:08