/sci/ - Science & Math

>>15085518
Hello anons, I'm working through a math textbook but I'm not sure about my answers. Can I post my solution to confirm whether it is correct as I work through a book? I'm working on problem 1a.
My solution:
Functional
Suppose [math] x_{1} = x_{2} \in X [/math], then [math] x_{1}\triangle_{X}y_{1} \implies y_{1} = x_{1} [/math] by definition of diagonal. The same argument can be apply to [math] x_{2} [/math] to obtain the following [math] y_{2} = x_{2} [/math].
Since [math] x_{1} = x_{2} \implies y_{1} = x_{1} = x_{2} = y_{2}[/math]. Thus, [math] \triangle_{X} [/math] is functional. Similar proof can be done for [math] \triangle_{Y} [/math].

Mutal Inverse Mapping
Suppose [math]R_{1}: X \to Y[/math] and [math]R_{2}: Y \to X[/math] such that [math] \triangle_{X} [/math] holds.
Meaning [math] \triangle_{X} = R_{2} \circ R_{1} = {(x,z}\in X^{2}| z = x} = e_{x}[/math] by definition.
Thus, [math] \triangle_{X} [/math] is mutally inverse mapping of X because [math] \triangle_{X} = R_{2} \circ R_{1} = e_{X}[/math].
We can apply the same argument for Y to conclude that the mutal inverse mapping of X and Y are [math]\triangle_{X}, \triangle_{Y} [/math].

-One question I have is, is second part is actually correct? Because I don't know the definition mutal inverse mapping on a set. The book only mention conditions for mutally inverse for two relations. So, I'm not entirely sure about that proof. Also, is the proof for function good enough? Thank you in advance.