Find the inverse of a function $f : X \rightarrow Y$ between two fuzzy topological spaces $X$ and $Y$?

Question

Suppose $(X, \tau_1), (Y, \tau_2)$ be two fuzzy topological spaces, where, $X=\{a\}, Y=\{x, y\}, \tau_1 =\{0_X, 1_X, \{(a, 0.3)\}\}, \tau_2 =\{0_Y, 1_Y, \{(x, 0.2), (y, 0.2)\}\}, $ and $f:X \rightarrow Y$ be a fuzzy function such that $f(a)=x$. Find the inverse of $\{(x, 0.5), (y, 0.3)\}$ in $X$. I think it will be $\{(a, 0.5)\}$ but than I am confuse about the inverse of the membership function of $y$ in this case.

Answer 1

What does it have to do with topology? You just want to find the inverse image of a fuzzy subset.

In the classical case, a subset $A$ of a set $Y$ can be seen as a map $_A:Y→\{0,1\}$. This is the characteristic function of $A$, which sends $x∈Y$ on $1$ if $x∈A$ and on $0$ otherwise. It takes an element of $Y$ and tells you whether it is in $A$ or not. To take the inverse image by $f:X→Y$ of a subset $_A:Y→\{0,1\}$, you just take the composite $_A ∘ f : X→\{0,1\}$. In other words, $_{f^{-1}(A)} = _A ∘ f$ (to know if $x∈f^{-1}(A)$, you look if $f(x)∈A$).

In the fuzzy case, this is the same: a fuzzy subset of $Y$ is a map $Y→[0,1]$ telling how much an element belongs to that subset. To know the inverse image of $_A:Y→[0,1]$ by $f:X→Y$, you just compose $f$ and $_A$ (to know how much $x∈f^{-1}(A)$, you look how much $f(x)∈A$).

Here, you compose the map $(a↦x)$ with the map $(x↦0.5; y↦0.3)$, and you find that $a$ is sent on $0.5$, so you are right.