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Given two sets $X$ and $Y$, and a function $f : X \to Y$, would it be correct to write: \begin{equation} \forall y \in \{a,b,c\} : \textit{Fy} \triangleq \{ x \in X \mid f(x) = y \}. \end{equation} as an equivalent of: \begin{equation} \textit{Fa} \triangleq \{ x \in X \mid f(x) = a \}.\\ \textit{Fb} \triangleq \{ x \in X \mid f(x) = b \}.\\ \textit{Fc} \triangleq \{ x \in X \mid f(x) = c \}.\\ \end{equation}

I'm basically looking for an equivalent of the evaluation function.

N.B. I don't want to write $F_y$ instead of $Fy$.

Thanks for you answers.

NS

N. Short
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1 Answers1

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On usually writes

$$f^{-1}[y] := \{x\in X\mid f(x)=y\}.$$

It is called the "preimage" (or "inverse image"). You can compute the preimage of a set as well, i.e.

$$f^{-1}[\tilde Y]:=\{x\in X\mid f(x)\in \tilde Y\}.$$

Then $f^{-1}[y]$ is equivalent to $f^{-1}[\{y\}]$. When it is known that the resulting set contains exactly a single element (when $f$ is bijective/one-to-one), then one often writes $f^{-1}(y)$ instead, and associates the result with this single element. This is called the "inverse function".

Note: Taste might differ and the use of square brackets and parantheses for the preimage is not standard. I prefer the notation presented here, because it is clear whether the result is a set or a single element.

M. Winter
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  • I had not seen $f^{-1}[y] $ for $f^{-1}[{y}]$, but with the square brackets there should be no confusion. As an aside, when I was first taught about this I was shown a nonstandard unambiguous notation $f^{<}(B)$ instead of $f^{-1}[B] $, so that $f^{<}:\mathbf{P}(Y)\to\mathbf{P}(X)$ was a genuine function. – Mark S. Mar 14 '18 at 15:28
  • Thank you for your clear answer. I will use this notation. However, I still wonder if there is a mathematical concise way to say, for example: "for each $x \in {a,b,c}$, we write $Fx$ instead of $F(x)$. Basically, I want to substitute $x$ in the name of the variable; hence my question about the existence of an "eval" function equivalent. – N. Short Mar 14 '18 at 15:51
  • @N.Short I am not familiar with any "eval"-functions, so can you explain a bit more what you need exactly? – M. Winter Mar 14 '18 at 15:53
  • An "eval" fct. (in certain programming languages) is a fct. that evaluates a string (an array of characters) as if it was code. eval("a=b+c") is the same as running a=b+c.

    I thought it was a good analogy with my problem, since what I want to do is find a math. concise way of writing: I want to substitute all $F(x)$ by $Fx$ where $x$ is the var. But, problem is my variable $x$ is embedded into the name of my vars.

    Simply said, I want to have var. names $Fa$, $Fb$, $Fc$ for $F(a)$, $F(b)$, $F(c)$, & explain this desire with a math formula that starts with: $\forall x \in {a,b,c}…$

     – N. Short Mar 14 '18 at 16:05
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    @N.Short I think you should explain a bit more about your use case. It might be that $\forall x\in{a,b,c}:Fx ...$ is totally fine, but it also seems that you are constructing a character string rather than a math formula, in which case you have to state which part of the string are characters and which are placeholders for the actual variable. – M. Winter Mar 14 '18 at 16:54
  • @M.Winter Yes, I am constructing a string that stands for a variable name. I want to replace some standard notations with more compact ones for the sake of readability. After some thought, I think it's better just to explain it with text as it would probably be too long/complicated to do it with a math formula (I wouldn't have answered my question though). – N. Short Mar 14 '18 at 17:01