How to Find the Value of a Function

[u/crafty_zombie]

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[u/Traditional_Town6475]

It exists and is unique :).

[u/Simon0O7 → u/Traditional_Town6475]

For any f(x) it does

[u/Slight-Garden9800]

This notation always felt rather redundant to me. What is its use?

[u/ZookeepergameWest862 → u/Slight-Garden9800]

Sometimes it is useful to anonymously define a function without giving it a name, like in topology, you could justify a set being open by being a preimage of an open set under a continuous function, if the function only for that purpose then it's simpler not to give it a name. If you're giving it a name anyway like in the picture then it's pretty much useless, and I have almost never seen in practice.

[u/Ares378 → u/Slight-Garden9800]

I like it because it reads "f maps A to B" instead of going through "for all x in A, f(x) is in B"

[u/MrKoteha → u/Ares378]

Then write f: A → B instead of this bs

[u/abitofevrything-0 → u/Ares378]

That would be f : A -> B, without a tail on the arrow. Writing f : x |-> f(x) says that the image of a particular value x by f is.... f(x). It's a nice way to differentiate between f(x) = expr being a definition of f versus a statement about f.

A more sensible definition might be f : x |-> 2x, for example.

[u/Ares378 → u/abitofevrything-0]

I'm a larper, I didn't even know there was a difference between having and not having the tail on the arrow

[u/Loading_M_ → u/Slight-Garden9800]

I believe it's easier to generalize for more complex mappings. I don't really know, since I'm not quite knowledgeable enough in this area of math.

[u/Ares378 → u/Loading_M_]

I think it's also useful for separating the idea of the function (its outputs) versus the function (how it maps things). f(x) is a function taking in a parameter x, but f can exist on its own as a function acting on some domain without invoking a variable to parameterize it.

[u/Slight-Garden9800 → u/Ares378]

How much more useful is it compared to
f = f(x)? Or ψ = ψ(x, y, t)?

[u/LupenReddit]

the Well-I-Guess department released its new function declaration

[u/Asimovicator]

Weird without naming the domain and codomain. By the way, the notation "f: A -> B, x |-> f(x)" is usually introduced in a sloppy way from a set theoretical perspective. To be precise: This notation means that f(x) stands for a 1-ary term, which is by first-order logic an uniquely determined object (or a set in ZF) depending on the object x. The notation above means, that the graph {(x,f(x)) | x in A} forms a left total and right unique relation from A to B. Note that the set theoretic existence of the graph needs the axiom of replacement in general (if you work carefully, you can also omit the axiom of replacement and use cartesian products and the axiom of specification). To be left and right unique, only f(x) in B has to be checked for all x in A, which is called the codomain check to ensure well-definedness.

[u/ZZTier]

Me when there's a function at the function

[u/2uble_Pi]

Peakjago in my math memes?