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I have a question, If we have $A=\{1\}$, Can I say it's commutative? it demands at least two different objects?

I think you can look at $(1,1)$ and say that $1+1$ is equal to $1+1$.


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Can you say what is commutative? – Git Gud Apr 19 '14 at 13:35
It means that a+b is equal to b+a. – user144228 Apr 19 '14 at 13:37
I think on a set with one element you can define one only (binary) operation and that ends to be commutative. – MattAllegro Apr 19 '14 at 13:40
@Git Gud: sorry master, but what you mean with $+$ isn't strictly sum of reals/integers? Isn't the result of any binary operation $+$ or $\ast$ on $\left\{a\right\}$ equals to $a+a=a$? – MattAllegro Apr 19 '14 at 13:42
Thanks everyone for your help! – user144228 Apr 19 '14 at 13:44

An set $A=\{x\}$ with one element has exactly one binary operation, the "identity" binary operation given by $x+x=x$. So yes, with this operation $A$ is commutative. Given any $a,b\in A$ we have $a=x,b=x$ so $a+b=x+x=b+a$.

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