R^2 R^3 problem.

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lollipop2046

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Suppose K is the xy-pane in R^3; that is, K ={(a1, a2, 0): a1, a2 from R}. Then K=R^2.

Why is this statment wrong? Is it because of the zero in the z-coordinate? But the coordinatewise addition / multiplication will only gives you back value on the xy-plane..

Please help!
 
What is the definition of R<sup>2</sup>? (I don't know what your book provides. Please look up the definition and reply.)

Eliz.
 
Why do you think that it is wrong?
It is certainly a MODEL for ℜ<SUP>2</SUP>.
But then the yz-plane is also a model for ℜ<SUP>2</SUP>.
You said “K=ℜ<SUP>2</SUP>”. Be careful with equality.
In mathematics, in most cases all we have is models.
 
I think R^2 just refers to the 2-D plane? Because the book didn't give any further information on the problem. So, I think it will just be the normal way we define it as the R^3 in the quesiton refers to the xyz plane..

I said the statement is wrong is because that's what it says at the back of the book...

pka: what do u mean by model? is it another term for subspaces?
 
or can R^2 refers to other planes if it didn't specify? But I thought a 2 dimensional space is a space of x and y....
 
lollipop2046 said:
or can R^2 refers to other planes if it didn't specify?
Click to expand...
I don't know. How does your book define R<sup>2</sup>? (Look up the definition, and reply with the full text.)

Eliz.
 
I've looked through the symbols page, index and the chapter where this question come from, they just didn't define R^2. And the question is just as it is, no further information.

So I think the reason that they claim this statement to be false is beacause they are not defining R^2 to be the cartesian 2D plane.

Thanks for replying!
 
“what do u mean by model? is it another term for subspaces?”
No, absolutely not.
A model is simply a set with operations that represents a system of axioms.
Moreover, there is NO xyz plane. Planes are two dimensional: they are ℜ<SUP>2</SUP>.
ℜ<SUP>3</SUP> is a set of 3-tuples with a set of axioms.
 
oops, sorry pka, I didn't see this line in your first post..

"But then the yz-plane is also a model for ℜ2. "

Thanks!
 
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