Mark44 said:
There certainly is ambiguity in the symbol as it is ordinarily used. Consider these two equations:
1. (x + 1)2 = x2 + 2x + 1
2. (x + 1)2 = 4
I didn't mean to imply that the "=" sign can't be ambiguous, I mean it is not the source of ambiguity here. In any case, the first is a true statement, the second is not a statement, and is wrong to write as a true statement. I could presume its truth by saying "let (x + 1)
2 = 4" but it is not a logical statement to write down precisely
because it has no inherent truth value.
The equals sign means the same thing in both cases.
You lost me here. I defined f and g as functions and then set them equal.
Yes, you defined ##f## and ##g## as functions, and they are functions, but one can use the symbol "##f(x)##" to either mean the function itself, or as an element of the range of the function, in different contexts.
By saying "then f(x) = g(x) if x = 2" you are saying that "the range element f(x) is equal to the range element g(x) whenever x = 2," you are not saying that the
functions ##f(x)## and ##g(x)## are equal whenever x = 2, you are saying that the functions map the domain element 2 to the same range element, 2. In other words, you are using f(x) and g(x) to refer to a specific member of the range corresponding to a specific member of the domain, x. They are not being used to represent the
functions f and g in this usage.
In other words, when I say "when x = 2, f(x) = 5" I am using the symbol "f(x)" to refer to an element of the range of the function f, I am not using it to refer to the function itself, I am not asserting that the
function f is equal to the number 5. Similarly, if I say "when x = 2, f(x) = 5 and g(x) = 5, so f(x) = g(x) when x = 2" I am using f(x) and g(x) to refer to elements of the ranges of f and g respectively. I am not saying that the
function f(x) is equal to the
function g(x) whenever x = 2.
But then you need to provide a formula for the function, which is what I did in my examples. I don't see that the notation with the arrow provides any clarity.
I don't know what you mean by needing to provide a formula with the function.
The notation with the arrow provides clarity because ##f: X \rightarrow Y## never refers to some element of the codomain, it always means the function itself. ##f(x)## may mean both, depending on the context.
Let ##f: X \rightarrow Y## be a function.
Then for any ##x \in X, f(x) \in Y##. I am not saying that the function f itself is a member of its own codomain.. "##f(x)##" does not refer to the function ##f## in this usage.
Have you ever seen anyone write something like:
Let ##f(x)## be a function.
Then for any ##x \in X, ( f: X \rightarrow Y ) \in Y##.
Because I sure haven't.
You're essentially arguing my point here. What I said was that we should distinguish between conditional equality of two expressions (each of which happen to be functions in my example) and unconditional or identical equality of the expressions.
Yes, but we can distinguish them automatically by simply clarifying what we are talking about. "Conditional equality of two functions" is just another way of saying "an instance where the output of the functions are equal." The
output of the functions, not the functions. So to write ##f(x) = g(x)##, and to mean "conditional equality of two functions" means you are referring to ##f(x)## and ##g(x)## as "outputs of the function" and not the functions.
If I say:
##f: X \rightarrow Y = g: X \rightarrow Y##
there is absolutely no way to interpret that as "conditional equality of two functions." It means the functions are equal, always.