Multivariable function( 2 dim to 2 dim) set theoretic analysis

Discussion in 'Analysis and Topology' started by deter_gustav01, Oct 20, 2023.

  1. deter_gustav01

    deter_gustav01

    Joined:
    Oct 20, 2023
    Messages:
    1
    Likes Received:
    0
    Hi there,

    I have this question below and it is about how to prove that the function is ONTO,
    With sets, which are arbitrary sets U, V, W, Y.

    ------------------------------------------------------------------------------------------
    Let f:U→V and g:W→Y be onto functions.

    Prove that ϕ:U×W→V×Y, defined by ϕ(u,w)=(f(u),g(w)), where
    ∀u∈U,∀w∈W,
    is an onto function.
    --------------------------------------------------------------------------------------

    My approach was to say let an element, coordinate pair (v,y) ∈ VxY,
    And since we are given from the beginning that f and g are ONTO,
    that allows one to write that f(u) = v and g(w) = y.
    ϕ(u,w)=(v,y).

    From this point, I am not certain how to create some
    more algebra, to somehow illustrate that ϕ:U×W→V×Y is an ONTO function.

    Hope someone can provide some insight into this.
     
    deter_gustav01, Oct 20, 2023
    #1
  2. deter_gustav01

    HallsofIvy

    Joined:
    Nov 6, 2021
    Messages:
    160
    Likes Received:
    78
    To prove that a function, f:X-> Y, s "ONTO" set Y we must show that for any y in Y, there exist x in X such that f(x)= y.

    In this case if y is in Vx Y then y= (v, y) with v in V and y in Y. Since f: U-> V is "onto" there exist u in U such that f(u)= v. Since g:W-> Y is "onto" there exist w in W such that g(w)= y. Then (u,w) is in UxW such that fxg(u, w)= (f(u), g(w))= (v, y) so f x g is "onto".
     
    HallsofIvy, Nov 20, 2023
    #2
  3. deter_gustav01

    ROBERTMILLS

    Joined:
    Feb 27, 2024
    Messages:
    3
    Likes Received:
    1
    To demonstrate that a function f:X→Y is "onto" or surjective onto set Y, we need to prove that for every y in set Y, there exists an x in set X such that f(x)=y.

    In this scenario, if y is an element in Y, it can be represented as (v,y) with v in V and y in Y. Given that f:U→V is "onto," there exists an u in U such that f(u)=v. Similarly, as g:W→Y is "onto," there exists a w in W such that g(w)=y. Thus, (u,w) belongs to U×W such that f×g(u,w)=(f(u),g(w))=(v,y), demonstrating that f×g is "onto".

    Many students find it difficult to do such types of question. Many students also miss their assignment due to not knowing the basic concepts. If you are one of them facing such difficulties, I would suggest you to visit Maths Assignment Help once. You can also contact them on +1 (315) 557-6473.
     
    ROBERTMILLS, Mar 12, 2024
    #3
Ask a Question

Want to reply to this thread or ask your own question?

You'll need to choose a username for the site, which only take a couple of moments (here). After that, you can post your question and our members will help you out.
Similar Threads
There are no similar threads yet.
Loading...