Create an injection from {a^2+b^2

  • Thread starter aaaa202
  • Start date
  • #1
1,170
3

Homework Statement


Is it possible to create an injection from {a^2+b^2 l a,b[itex]\in[/itex]Q} to {{a+b l a,b[itex]\in[/itex]Q}

Homework Equations


I am not sure about this. The problem is that many combinations of a,b can yield the same number, so how do I tackle this obstacle?


The Attempt at a Solution

 

Answers and Replies

  • #2
UltrafastPED
Science Advisor
Gold Member
1,912
216
Consider the definition of injection.
 
  • #3
1,170
3
the definition says:

x1≠x2 => f(x1)≠f(x2)

But the problem is as said that many combinations give rise to the same numbers in the set.
 
  • #4
UltrafastPED
Science Advisor
Gold Member
1,912
216
Then perhaps it is not possible!
 
  • #5
1,170
3
it certainly should be, since the two sets have same cardinality. I just need help with coming up with a well defined function that does it. Problem is you can't really say something like f(a^2+b^2)=lal+lbl because of the earlier stated.
 
  • #6
pbuk
Science Advisor
Gold Member
2,048
817
it certainly should be, since the two sets have same cardinality.
Is this necessarily true?

I just need help with coming up with a well defined function that does it.
How about x → x?
 
  • #7
jbunniii
Science Advisor
Homework Helper
Insights Author
Gold Member
3,470
246
It might simplify the problem somewhat to observe that ##\{a+b \, | \, a,b\in \mathbb{Q}\}## is simply ##\mathbb{Q}##.
 
  • #8
pbuk
Science Advisor
Gold Member
2,048
817
I've just looked at a couple of your other posts like this one where again you seem to be confused about what you mean by {a^2+b^2 l a,b∈Q}. Can you see that all elements of this set are in ## \mathbb{Q} ##, and that a and b are only relevant in selecting which elements of ## \mathbb{Q} ## are elements of this set? You can't then define a function f(a,b) and expect it to have any meaning in relation to this set without defining how you get the unique pair (a,b) from ## x \in \mathbb{Q} ##.

Before you try to think about jections (I hate that word) you need to think more clearly about the membership of sets and the domain, co-domain and range of relations between them.
 
Last edited:

Related Threads on Create an injection from {a^2+b^2

Replies
4
Views
1K
Replies
11
Views
8K
Replies
1
Views
3K
Replies
36
Views
896
Replies
6
Views
1K
Replies
8
Views
1K
  • Last Post
Replies
5
Views
1K
  • Last Post
Replies
1
Views
660
  • Last Post
Replies
2
Views
1K
Top