Def.1. A binary operation on a nonempty set $A$ is a map $f:A\times A\to A$ such that
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$f$ is defined for every pair of elements in $A$, and
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$f$ uniquely associates each pair of elements in $A$ to some element of $A$.
It seems that the definition of a binary operation includes the definition of a well defined operation (condition 2). But I see other books using the term: Well defined binary operation ! Is there a not well defined binary operation ?
So that just confuses me. What is meant by that ?
Best Answer
Any binary operation is well defined. An author may write "well defined binary operation" when the fact that it is well defined is not obvious. If it weren't, it wouldn't be a binary operation, or even a thing.
Well defined usually means what you said: if $a=a'$ and $b=b'$, then $a*b=a'*b'$. One time when this is not obvious is when rather than using actual equality you're using an equivalence relation and defining the binary operation on elements of the equivalence classes. Then it is well defined on the equivalence classes if it does not depend on the choice of representative. This is the most common case I've seen where the author feels the need to emphasize that it is well defined.