[Cardinality] Prove there is no bijection between two sets

In summary, the conversation discusses how to prove that there is no continuous bijection from the unit circle to R. It is suggested to use cardinality to show this, since two sets with different cardinality cannot have a bijection between them. It is also mentioned that the unit circle and R have different cardinalities, with R having the cardinality of continuum. The conversation also briefly touches on the concept of compactness and whether a continuous map can exist from a compact set to a non-compact set. Ultimately, it is concluded that there is no continuous bijection from the unit circle to R.
  • #1
mick25
13
0

Homework Statement


prove there is no continuous bijection from the unit circle (the boundary; x^2+y^2=1) to R


Homework Equations





The Attempt at a Solution



is this possible to show by cardinality? since if two sets have different cardinality, then there is no bijection between those two sets

R has the cardinality of continuum

the unit circle is defined on [-1,1]x[-1,1] and since [a,b] has same cardinality as R for all a,b, cardinality of the unit circle would be c*c = c^2 but c^2=c, but this can't be since then there would be a bijection between the unit circle and R

?
 
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  • #2
mick25 said:

Homework Statement


prove there is no continuous bijection from the unit circle (the boundary; x^2+y^2=1) to R


Homework Equations





The Attempt at a Solution



is this possible to show by cardinality? since if two sets have different cardinality, then there is no bijection between those two sets

R has the cardinality of continuum

the unit circle is defined on [-1,1]x[-1,1] and since [a,b] has same cardinality as R for all a,b, cardinality of the unit circle would be c*c = c^2 but c^2=c, but this can't be since then there would be a bijection between the unit circle and R

?

How about something easier?

Is R compact? Is the unit circle compact?

Can you have continuous map from a compact set to a non compact set?
 
  • #3
fauboca said:
How about something easier?

Is R compact? Is the unit circle compact?

Can you have continuous map from a compact set to a non compact set?

nope

i just realized it after posting this thread but i don't know how to delete it now

thanks
 

1. How do you prove that there is no bijection between two sets?

To prove that there is no bijection between two sets, you must demonstrate that there is either no function that maps one set onto the other, or that there is a function that violates the definition of a bijection.

2. What is a bijection?

A bijection is a function that is both injective (one-to-one) and surjective (onto). This means that each element in the domain is paired with exactly one element in the codomain, and every element in the codomain is mapped to by at least one element in the domain.

3. Why is it important to prove that there is no bijection between two sets?

Proving that there is no bijection between two sets is important because it helps us understand the relationship between the two sets. It also allows us to determine if the sets have the same cardinality, which is useful in many mathematical and scientific contexts.

4. What are some methods for proving that there is no bijection between two sets?

There are several methods for proving that there is no bijection between two sets. One approach is to assume that there is a bijection and then show that it leads to a contradiction. Another method is to use cardinality rules and properties to show that the two sets have different cardinalities. Additionally, you can use specific examples or counterexamples to demonstrate that there is no bijection between the two sets.

5. Can there be a bijection between two infinite sets?

Yes, there can be a bijection between two infinite sets. For example, the sets of positive integers and even integers have the same cardinality, even though the set of positive integers is a proper subset of the set of even integers. This is because there is a bijection between the two sets (e.g. the function f(x) = 2x). However, there are also infinite sets that do not have the same cardinality and therefore cannot have a bijection between them (e.g. the sets of positive integers and real numbers).

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