About the definition of binary operation on a set, in my notes it says a binary operation on S is a map $*:S\times S\to S$, it does not have to be a function, it is a mapping. But in the textbook, it says while defining a binary operation on a set $S$, we must be sure that exactly one element is assigned to each possible ordered pair of elements. Doesn't that mean it is a function? I am confused here, does a binary operation have to be a function? Thanks

  • $\begingroup$ Usually, map (mapping) and function mean exactly the same. How do they differ in your notes? $\endgroup$ – Tobias Kildetoft Mar 11 '13 at 17:32
  • $\begingroup$ in my notes, it explicitly says that it is just a map, not a function. What i understand from this sentence is, more than one element can be assigned to a pair $\endgroup$ – Yasin Razlık Mar 11 '13 at 17:33
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    $\begingroup$ We can’t answer the question until we know exactly how the terms mapping and function have been defined for you. In most contexts either they mean exactly the same thing, or mappings are functions with some additional property; in neither case is it possible to have a mapping that is not a function. $\endgroup$ – Brian M. Scott Mar 11 '13 at 17:35
  • $\begingroup$ You still haven't told us why you think a mapping and a function are different things. $\endgroup$ – Thomas Andrews Mar 11 '13 at 17:36
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    $\begingroup$ That’s wrong by any definition of mapping that I’ve ever encountered. $\endgroup$ – Brian M. Scott Mar 11 '13 at 17:39

A binary operation on $S$ is a function $S\times S\to S$. You are correct.

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