A binary operator in mathematics is defined as an operator defined on a set that takes two elements of the set and returns a single element. An example would be integer multiplication "*" where a,b are both integers and a*b returns an integer.

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''Does it have to return an element of the same set?''

No. Division is a good example: division of two integers may yield a fraction (which is obviously not of the set of integers). This does not keep division from being an operator; it merely means that the set of integers is not closed over division.

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''What do you call an operator that takes one argument and returns two elements?  e.g.:  SQRT(4) -> {-2, 2}?''

That would be "unary" because of the number of arguments it takes. I don't know if there are different names according to the number of elements returned.

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CategoryMath