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This is probably something very obvious, but I am a little confused. It's about the associative law.

It is known that a binary structure $(S, *)$ is associative if:

$(a * b) *c = a * (b * c)$ for all elements in $S$. It's also closed, as well.

Well, say I wanted to do this with for elements in the set: $\{a,b,c,d\}$. Would parenthesizing for the associative law go something like this?

$(a * b) * c * d = a * b * (c * d) = a * (b * c) * d$ .....etc.

I know the gist of the law states that it doesn't matter which two elements you enact the operation with first, it will always be the same. But for understanding and a hw exercise, I need to know if I'm going about this correctly.

Ken
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user121615
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    $(ab)cd$ is ambiguous. $$ is binary, so you need to know, for each $$, what are the two operands. So $((ab)c)d$ is one and $(ab)(cd)$ is the other. Again, if $$ is associative, they are equal, which you can prove from the basic associative law. – Thomas Andrews Feb 25 '15 at 20:33
  • Also, so is $a\ast (b\ast (c\ast d))$ – Graham Kemp Feb 25 '15 at 20:35
  • When you say ambiguous in the mathematical sense, what exactly does that mean? Are you saying it isn't very clear what things are supposed to be multiplied when I write (a * b) * c * d? – user121615 Feb 25 '15 at 20:36
  • Not multiplied, I meant binary operation – user121615 Feb 25 '15 at 20:37
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    oooooh Wait, nevermind. I get it! Thanks – user121615 Feb 25 '15 at 20:38
  • @GrahamKemp Unclear if you mean that 'also' in reference to my comment, but your comment is not one of the ambiguous possibilities for the meaning of $(ab)c*d$. – Thomas Andrews Feb 25 '15 at 20:48

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