Question

General question about maps.

Suppose I have a function \(f:X\times X\rightarrow Y\times Y\). Is there a name for the function \(g:X\rightarrow Y\) where \(g(x) \) is first coordinate in \(f((x,y))\)

Answer 1

wait let me reword that

Answer 2

Okay.

Answer 3

Hmm let me think...

Answer 4

not quite a restriction and note quite a projection lol

Answer 5

resjection?

Answer 6

It can't be projection for a sense, don't you agree?? But it kinda more of restriction hmmm

Answer 7

yeah, no idea... Seems like this would be something done many times so I thought there would be a good name....

Answer 8

Lol... totally agree.

Answer 9

Looks relevant https://en.wikipedia.org/wiki/Functional_decomposition

Answer 10

I think it's "projection". Like this: \(\pi_1(x,y) = x\). Then \(g(x) = \pi_1(f(x,y))\).
While restriction is used as in: \(2\in\mathbb R\) -> \(2\in \mathbb N\) (same object, but belongs to a smaller set).

Answer 11

How is that not a projection?

Answer 12

I dont know. I thought projection would be XxY to X

Answer 13

maybe it is

Answer 14

I see what you mean now. It is a projection of \(f\).

Quoted from Wikipedia:

"In set theory:
An operation typified by the j-th projection map, written \(\operatorname{proj_j}\), that takes an element \(x = (x_1, \dots, x_j , \dots, x_k)\) of the cartesian product \(X_1 \times\dots\times X_j \times\dots\times X_k\) to the value \(\operatorname{proj_j}(x) = x_j\). This map is always surjective."