Problem:

Let S be a set having exactly one element. How many different binary operations can be defined on S? Answer the question is S has exactly 2 elements, exactly 3 elements, exactly n elements.

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I think I'm getting a little confused with the term "element", because it seems to me, there can be an infinite amount of binary operations performed on a set that's simply non-empty.

Can't I just say

and then start listing off different things the binary operation

could be? (cos, sin, +, -, +7, literally an infinite amount of things)

Or does this problem mean element as in the arbitrary a, not being operated on with itself, and it just wants me to count the different combinations I can use the elements with? So if n=1, there are zero binary operations (or 1 if I can do a*a?). With n=2, I could do a*b, b*a, (a*a), (b*b)?

Any clarification appreciated.