Hello, this is for my Advanced Calculus Class.
If A, B, and C are sets, then prove...
a) A ~ A
b) A ~ B if and only if B ~ A
c) A ~ B and B ~ C implies A ~ C
I need to prove each one, I tried working on each but I just end up getting lost or messing it up and it doesn't make sense. Please prove them.
I added items in parentheses to above quote.
Shilov uses to denote equivalence, and defines equivalence to be a one-to-one correspondence between sets. In that case, symmetry and reflexivity are obvious, and for transitivity:
Corresponding to a in A there is a unique b in B and corresponding to b in B there is a unique c in C so corresponding to a in A there is a unique c in C. And then the other way around.
From Wolfram Math World (google "injection"):
"In other words, is an injection if it maps distinct objects to distinct objects. An injection is sometimes also called one-to-one."
However, the conditions a), b), and c) of OP can only be satisfied if it is one-to-one-onto, which I used in my proof, ie, we are still talking about the same thing, one-to-one onto.
Speaking of my original proof, I identified a) and b) of OP as symmetry and reflexivity, when they are conventionally identified the other way around, except by Shilov (pg 30), which I happened to have at hand and because I can't remember which is which and looks symmetric to me- I guess it makes more sense that way to us Russians. Technically, and from a purist point of view, having defined my terms my proof was correct within the context of the definition, and, btw, also for the reverse definition, and it really didn't depend on how I happened to identify conditions a) and b) anyhow. Its a triviality which produced a very harsh response from Plato, and a warning message from the powers that be.