If functions are a mapping of values to other values, why can't I subtract two functions to get the difference in domains? egf1(x) = {1 -> 2, 2 -> 3, 3 -> 4}f2(x) = {1 -> 2}f1 - f2 = {2 -> 3, 3 -> 4}
>>16787280>why can't IWho's stopping you?
>>16787280functions are defined from one set to anotherthere is no general notion of "subtraction" of functions. when you can subtract functions, it's because their codomains are the same and have subtraction defined on them.search function image and preimage, these are the kinds of ideas you are thinking about.
>>16787280In mathematics you can define many operations. The only question is if this operation is useful for anything.
>>16787299being able to algebraically manipulate the internals of functions is fundamentally useful since it lets you write shorter proofs. rather than defining a new function from scratch, just change another function you already have defined.imagine being able to define the inverse of any arbitrary function, and then restrict the domain of the first function to match-up with the second function. eg anything that might result in division by zero. you could do all of that by hand, or you could just algebraically manipulate functions and save yourself weeks of your life.
>>16787344>since it lets you write shorter proofsThat's what a maid does, she cleans a house. That's not a math thing, it's a housekeeper thing.
>>16787344But you still have to do the calculations to get anything useful out of it.Lambda Calculus does define a useful and usable definition of applying the subtraction function to 2 function arguments but then the difficulty becomes writings the 2 functions using lambda calculus
>>16787358>Lambda CalculusI'm a retard with a buzzword in my pocket
>>16787344>being able to algebraically manipulate the internals of functions is fundamentally useful since it lets you write shorter proofs.Provide an example.
>>16787280you’re basically subtracting 2 small turing machines so you’re going to have to figure out how that would work
>>16787392>you’re basically subtracting 2 small turing machinesThat's wrong, though.
>>16787280that's just set subtraction between functions encoded as relationsyou are essentially restricting the domain of function f1
>>16787405no its not libtard
>>16787280Depending on the qualifiers and restrictions, there are some operations you can perform "on functions." The calculus of variations is all about finding "the best" smooth function to optimize some integral/integro-differential equation.
>>16787280Actually you can do this, exactly in the way that you describedBut doing this for e.g. a function over the real numbers is just not practical, so instead we subtract the bodies of functions that we know have the same codomains
>>16787344>imagine being able to defineImagine actually doing the work instead of asking others to imagine you did the work.
>>16787280>>16787457/threadOP no one is stopping you making up algebraic operations but it's unlikely to be original. If you want a name for this it's set subtraction obviously, denoted f1\f2.
