Suppose X, Y, Z are sets. If X ~ Y and Y ~ Z, prove that X ~ Z.

I thought we already proved this in my discrete math class, but I couldn't find it in my notes

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Okay my work on the proof so far is:

Suppose X, Y, Z are sets. Let X ~ Y and Y ~ Z. By equivalence, there are functions f and g such that f: X → Y where f is 1-1 and onto, and g: Y → Z where g is 1-1 and onto.

So now I have to show that there is a function h: X → Z that is 1-1 and onto. This is where I got stuck. Like for showing 1-1, I need to assume two elements (x_1, x_2) are equal and then show that when I plug them in function h, I get the same answer. For showing onto, I pick an arbitrary element z in Z and get an element in X that maps to z.

My thinking/scratch work:

Since f is 1-1, then when x_1 = x_2 then f(x_1) = f(x_2). f's range is Y and so f(x_1) and f(x_2) are in Y and so those are also equal since g is 1-1. So somehow it's tying that all together to show that h is 1-1. How can I do that clearly?

Now for onto, since f is onto, there is a y in Y such that there is an x in X that maps to y. I can use that 'y' in Y to map to a z in Z since g is onto. So can I then say that since x maps to y then x maps to z?

I hope my work isn't too confusing.

Thanks for your time.