Mapping onto a cross product

>>8401176
off yourself my dude

there's being a brainlet and then there's being a fucking retard.

>>8401184
does that mean you dont know the answer?

>>8401189
That's what it means.

>>8401189
you are calling the cartesian product of two sets a cross product. From this I deduced it wasn't worth my time answering your question

>>8401189
Really, though, I can't figure out what this notation means. if X mapped onto Y means that for every x value there is a corresponding y value, and if the cartesian plane can be represented by the cross product r*r

does z mapped onto z*z

mean that for every integer x there... what? i get lost here. how does this statement translate into english

>>8401194
This is the terminology used in Essentials of Discrete Mathematics, 3rd edition and in my lecture.

I've also found wolframalpha using the terminology interchangeably:

"The Cartesian product of two sets A and B (also called the product set, set direct product, or cross product)"

- http://mathworld.wolfram.com/CartesianProduct.html

I don't, however, want to turn this into a war of semantics. If you disagreed with my use of words you could've chosen to correct them and then explain the concept, but instead you decided to be a jerk. are you, by any chance, a CS major?

>>8401207
You can also see wolfram listening r*r*r as a three dimensional plane, but i still cant find anyone using this notation outside of my textbook, so again, if anyone could explain the notation in the OP, it would be greatly appreciated.

>>8401176

Stop using the term cross product for the time being. Rather, it is Z being mapped into Z^2, or as it is written here, Z x Z.

Here is another function that may make things more clear: f(x) = (x, x). This is R -> R^2

Also, look here:
https://en.wikipedia.org/wiki/Surjective_function
https://en.wikipedia.org/wiki/Cartesian_product

This may confuse you more, but for further understanding:
https://en.wikipedia.org/wiki/Vector-valued_function

File: compgirl.jpg (7KB, 300x168px) Image search: [Google]

7KB, 300x168px

f is obviously one-one. Check hat by taking f(a) and f(b) and equating them.

And to check its surjectivity just check whether (odd number - 3)/2 generates the set of integers or not.

OP, get the fuck off Mongolian Basketweaving forums and study. This shit is extremely basic and crucial.

>>8401224
>https://en.wikipedia.org/wiki/Vector-valued_function
He's in a module, not a vector space, don't introduce unnecessary definitions.

>>8401227
I understand how to prove its one-to-one and and that it's not onto, i just dont understand what exactly im looking at

for reference, i proved one to one by treating 2x+3 and x-4 as separate functions and set them equal to themselves using a and b
2(a)+3 = 2(b)+3
a=b
proof by contrapositive, etc.

proved not onto by setting 2x+3=0
x=-3/2 which is rational so not an integer

but,
i just dont understand what it meant by this notation z maps onto z*z
what does the graph of this problem look like?
z maps onto z^2 unfortunately doesn't really clarify it for me.

>>8401244
to further clarify, i understand that

z maps onto z*z where f(x) = (3x+2. x-4)
is equivalent to saying
f(z)=(3z+2, z-4)

I'm assuming you're confused by what it's asking in part (b). It's actually using some shit tier but common terminology.

In better terminology one would say it is asking if the function is "well defined".

However, from a more formal (clearer) perspective it's actually asking two questions.
>Given any value, z, in the domain:
>1) is f(z) defined?
>2) is f(z) deterministic?

By deterministic it's asking if it's unambiguous, or in other words, if given some value z, does f(z) evaluate to at most one thing? For instance, consider the square root. If I tried to define my own square root function and said sqrt(9)=3 and sqrt(9)=-3 then the function would be indeterministic/ambiguous. Such a function would not be well defined. An observant reader here will realize that these two properties are actually analogues of injective and surjective but in the opposite direction.

Moreover, while there are parts of mathematics that refer to such functions as partial functions or indeterministic functions, in mainstream mathematics the "definition of a function" is that it is a "well defined function", also the word "map" is sometimes used as a synonym for function (as in the OP picture).

Sooooo, what the OP picture is asking is if f is a function which is a roundabout way of asking if it's well defined.

>>8401233

My apologies, you are correct.

>>8401195

Consider f:X->(Y*Z)
This means that for each x in X, f(x)=(y,z) is an ordered pair in Y*Z, where y is in Y and z is in Z.

For instance, in the OP picture you have
f:0->(0,0)
or in other words f(0)=(0,0).

I think I perhaps may have gotten caught up in my own misinformed ideas and missed something obvious meant by this notation

Read in english z maps onto z*z
where f(x) is (2x+3, x-4)

is

z maps onto the set (2x+3. x-4)?

>>8401233
>le pretentious cancer
lul why are you assuming that much the most we should assume is that he's dealing with the product in the category of sets. Am I being profound yet?

>>8401258
Thank you

this is exactly what i was looking for

>>8401244
>what does the graph of this problem look like?

First note that each point on a graph is actually an ordered pair. Typically denoted as an x-coordinate and a y-coordinate, written (x,y).

So, to visualize this observe two separate things. A timeline, and a 2-dimensional graph. The input values for your function will come from your timeline and your output values will be displayed on the graph.

So, for instance, at time 0 you are at point (3,-4).
At time 1 you are at (5,-3).
and so on..

Depending on how well behaved your functions are (continuous etc..) they may draw out a nice line on your 2D-plane or maybe just a bunch of points. At any rate, observe that this explanation is consistent with one's intuition of taking 1 dimensional space and mapping it onto a 2-dimensional space.

>>8401278
thank you for this eloquent explanation.