On Monday, September 30, 2013 7:20:26 PM UTC-5, Hetware wrote:
> A function is a mapping from elements of a range to elements of a > > domain. Often the image is required to be single valued, but Thomas > > does not stipulate that requirement.
This "definition" does not agree with your prior assertion that a function is a set of ordered pairs; nor does your aside about the possibility of images being more than "single valued".
In any case, I'll wager that Thomas *does* specify that functions must yield a unique output given any particular input (or, if he defines functions as you did before, as sets of ordered pairs satisfying special conditions, then one of the conditions required of functions defined this way is that if (a,b) and (a,b') are in f, then b=b'; i.e., that they be single valued).
> If I can deterministically > > interpret a formal expression as such a mapping, then my interpretation > > of that formal expression satisfies the definition of a function.
But your *interpretation* of the expression is not necessarily the *intended* meaning of the expression.
As I said before, if functions are set of ordered pairs (or if functions are rules assigning to every valid input a corresponding output), then an expression is not a function. The expression in question is supposed to determine a function *in a particular, agreed upon manner*.
While you are free to set up your own particular and personal conventions, you are *not* free to impose those on the book you are reading, which no doubt established its own intended meaning before and that you are ignoring.