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**EliteAndoy** First off, thanks everyone for the replies, I really appreciate it. Anyways, if you were asking if "S has cardinality m", "|S| = m" and "S has m elements" all mean the same thing in our course, then yes it does. As for the claim, It's kinda obvious that |S|=|{1, 2, ..., m}|, but I can't really just say that it's obvious in this homework, can I? Plus I'm asked to show their one-to-one correspondence and so I'm just using those terms as a tool to resort to the "definition" of cardinality. I'm not saying that the definition is not true, I'm just saying that if it is untrue then my whole argument is also untrue.