Equivalence of sets proof assistance

[Agent M27]

Homework Statement

Suppose there exist three functions:
f:A$$\stackrel{1-1}{\rightarrow}$$B

g:B$$\stackrel{1-1}{\rightarrow}$$C

h:C$$\stackrel{1-1}{\rightarrow}$$A

Prove A$$\approx$$B$$\approx$$C

Do not assume the functions map onto their codomains.

Homework Equations

The Attempt at a Solution

I took a screenshot of my work and have attached it to the post. My question is concerning my last step, setting t=g$$\circ$$f. Mathematically speaking I don't know if it is legal, however I don't see why I would not be able to do such an operation. Thanks in advance for the assistance.

Joe

 

[Dick]

That's fine. But there's also no reason you can't just use transitivity to show A<=C just like you did for B without using the composition.

 

[Agent M27]

I didn't even think about the transitivity of A and C until you pointed it out! I always over complicate these things. I think I will employ that method tomorrow should this problem appear on my exam. Thanks for your help.

Joe