When two functions are equal?

[Anonymous No.8824512]

A function is a relation.
Let:

[math]f_{1}\subset \{0\}\times\{0\}=\{(0,0)\}[/math] or [math]f_{1}:\{0\}\rightarrow \{0\}[/math].
[math]f_{2}\subset \{0\}\times\{0,1\}=\{(0,0),(0,1)\}[/math] or [math]f_{2}:\{0\}\rightarrow \{0,1\}[/math].
[math]f_{1}(0)=0[/math] so [math]f_{1}=\{(0,0)\}[/math].
[math]f_{2}(0)=0[/math] so [math]f_{2}=\{(0,0)\}[/math].

So the the functions are equal despite being "defined" on different sets.
Can anyone clarify this?

[Anonymous No.8824515]

[Anonymous No.8824528]

if they're defined on different sets they can't be equal

[Anonymous No.8824538]

in particular f_1 is surjective and f_2 isn't

[Anonymous No.8824541]

https://math.stackexchange.com/questions/1403122/when-do-two-functions-become-equal

[Anonymous No.8824547]

>>8824512
In general, a function is not the same as a relation: Your [math] f_2 [/math] is not a function. For a function [math]f:A \to B[/math] it is not allowed that there are more then one [math] y \in B [/math] such that [math] f(x) = y[/math] for a specific [math] x [/math].

[Anonymous No.8824548]

>>8824547
> Your f2 is not a function.
???
his f2 is just f(0)=0
how is that not a function?

[Anonymous No.8824549]

>>8824547
>Your f2 is not a function.
It is.

[Anonymous No.8824552]

>>8824538
Makes sense.
>>8824541
Read part of it.
Thanks for the reference.

[Anonymous No.8824553]

There is a seperate long Wikipedia entry on the History of functions
https://en.wikipedia.org/wiki/History_of_the_function_concept

With [math] f_1 = \{ (0,0) \} [/math] you say [math] f_1 [/math] is a set.

A relation, i.e. a set of one or more pairs, that has the "function property" gives you a model of a function in set theory.
(The function property is that for each left entry of a pair, the "input", there are not several possible right entries, i.e. there is a unique "output".)

To make sense of a statement like "the function is surjective/onto", you need to define a function as a tuple or a triple. The first entry is the above relation, the others are the domain and codomain.
E.g. the function defined via g(x):=x^2 on R is surjective if you consider the codomain to be the positive real numbers R_+, but it's not surjective if you consider the codomain to be all of R.
Two functions, if you consider the codomain to be part of the data, disagree as soon as any of the entries disagree.

In type theory the type actually comes first and you can't even define a function without domain and codomains.

[Anonymous No.8824559]

>>8824549
>>8824548
Ups sorry, that was posted too fast.

[Anonymous No.8824631]

>>8824553
Nice, thanks.

[Anonymous No.8824655]

>>8824512
>https://math.stackexchange.com/questions/1403122/when-do-two-functions-become-equal
yes relation that maps one X from A to at most 1 Y from B... do you understand, piggot?

[Anonymous No.8824679]

>>8824512
you stupid pig, watch me do some magic

lleft hand = PIG
right hand = FAGGOT

magic-clap! PIGGOT
you're a fucking PIGGOT! worse than a faggot and a pig combined!

note the fact that 'C' means a strict subset it means that if ACB B will have some things that A won't have. a Function can't be C into AxB because if that happens, if cartesian product happens then it will link multiple Xs to the same Ys, impossible my friend


this is why, you stupid piggot, this is why functions only take x to one y, at most

bijective: every x has a Y, every Y has an X, but no more, no less, piggot

[Anonymous No.8824709]

>>8824512
Depending on your field of study, two functions are equal only when their domain and codomain are equal and they do the same actions on the sets. Different sets means the functions are not equal.