Expert: Paul Klarreich - 9/29/2009

Question
This question is from the section on binary operations.  It asks that if the set S has exactly one element in it, how many binary operations, #, can be performed on the set.  It then asks the same for S having 2, 3, and n elements.  I know they're all done in the same manner, but I just can't get started on the first one.  My initial thought was for |S|=1 there would be no binary operations, but I was only thinking of the standard operations.  I then tried to write down all the ones I can think of.

If S = {a}
Then SxS = {(a,a)}
1.) a # a = a

That was as far as I got.  I'm fairly certain the book doesn't want me to list all the binary operations, but merely come to some generalization.  I just don't know where to go from here.  Any help would be appreciated.

Answer
Questioner: Adam
Private: no
Subject:  
Question: This question is from the section on binary operations.  It asks that if the set S has exactly one element in it, how many binary operations, #, can be performed on the set.  It then asks the same for S having 2, 3, and n elements.  I know they're all done in the same manner, but I just can't get started on the first one.  My initial thought was for |S|=1 there would be no binary operations, but I was only thinking of the standard operations.  I then tried to write down all the ones I can think of.

If S = {a}
Then SxS = {(a,a)}
1.) a # a = a

That was as far as I got.  I'm fairly certain the book doesn't want me to list all the binary operations, but merely come to some generalization.  I just don't know where to go from here.  Any help would be appreciated
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I am not totally sure what is going on here, but it seems that:

1. Your # is not necessarily commutative.
2. Each 'binary' is actually a triple, since you have to get a 'product', too.

So, for example, if you have:

S = {a,b}, then you can have:

{a,b,a}, meaning  a # b = a
{a,b,b}, meaning  a # b = b

etc.

Therefore, if |S| = n, you should have  n^3 possible triples.

Does this make any sense?