Showing that two intervals have the same cardinality ?

In summary, the conversation discusses finding a bijective function to show that [0,1] and [0,2] have the same cardinality. It is mentioned that there are multiple functions that can fit this description and any of them can be used. The speaker is unsure about the process and asks if there is a specific formula to find the function. The response is that there is no formula, but intuition and experience are helpful.
  • #1
SMA_01
218
0

Homework Statement



I need to show that [0,1] and [0,2] have the same cardinality by giving a formula for a function that is bijective. Aren't there a number of functions that can fit this description? Can I then use any one? I'm a little confused, my teacher didn't really elaborate much upon this. Is there a specific formula to find the function, not only in this case, but generally?


Edit: I have used y=f(x)=2x, I'm guessing this process is pretty systematic when dealing with such intervals?
 
Last edited:
Physics news on Phys.org
  • #2
SMA_01 said:

Homework Statement



I need to show that [0,1] and [0,2] have the same cardinality by giving a formula for a function that is bijective. Aren't there a number of functions that can fit this description? Can I then use any one? I'm a little confused, my teacher didn't really elaborate much upon this. Is there a specific formula to find the function, not only in this case, but generally?Edit: I have used y=f(x)=2x, I'm guessing this process is pretty systematic when dealing with such intervals?

You can use any bijection. f(x) = 2x is fine. So is a non-linear one like f(x) = x(3-x), which is bijective over [0,1]. It's your choice. But the former has the advantage of being immediately and intuitively obvious.
 
  • #3
SMA_01 said:

Homework Statement


Is there a specific formula to find the function, not only in this case, but generally?

There's definitely no "formula" here. Intuition and experience are your best friends when seeking for Bijections.
 

Related to Showing that two intervals have the same cardinality ?

1. How do you prove that two intervals have the same cardinality?

To prove that two intervals have the same cardinality, you can use the method of bijection. This means finding a one-to-one correspondence between the elements of the two intervals. If such a correspondence can be established, then the intervals have the same cardinality.

2. What is the definition of cardinality?

Cardinality is a measure of the number of elements in a set. It is often denoted by the symbol |S|, where S is the set. It can also be thought of as the size or quantity of a set.

3. Can two intervals with different lengths have the same cardinality?

Yes, two intervals with different lengths can have the same cardinality. For example, the interval [0,1] and the interval [0,2] both have an infinite number of elements and can be put into a one-to-one correspondence, proving that they have the same cardinality.

4. What is the importance of showing that two intervals have the same cardinality?

Showing that two intervals have the same cardinality is important because it allows us to compare the sizes of different sets. It is a fundamental concept in mathematics and is used to understand the concept of infinity and the relationship between different infinite sets.

5. Are there any other methods to show that two intervals have the same cardinality besides bijection?

Yes, there are other methods to show that two intervals have the same cardinality. These include using the methods of injection and surjection. Injection involves finding a one-to-one function from one interval to the other, while surjection involves finding a function that maps elements from the first interval onto every element in the second interval. If either of these methods can be successfully applied, then the intervals have the same cardinality.

Similar threads

Cardinality using equivalences
    • Calculus and Beyond Homework Help
Replies
1
Views
1K
I [0,1] same cardinality as (0,1)
    • General Math
Replies
10
Views
3K
Do I need a function to show a bijection between intervals?
    • Calculus and Beyond Homework Help
Replies
16
Views
6K
Every subset of a finite set is finite
    • Calculus and Beyond Homework Help
Replies
1
Views
1K
Is there a way to prove equivalent cardinalites?
    • Calculus and Beyond Homework Help
Replies
1
Views
2K
Show intervals of real numbers have the same cardinality
    • Calculus and Beyond Homework Help
Replies
6
Views
5K
Discrete math - Infinite sets having the same cardinality.
    • Calculus and Beyond Homework Help
Replies
1
Views
2K
I Collection of finite unions of half-open intervals form an algebra
    • Topology and Analysis
Replies
3
Views
311
Confidence Interval based on two sample tests
    • Calculus and Beyond Homework Help
Replies
21
Views
2K
Proof that an interval is a confidence interval for Geom(q)
    • Calculus and Beyond Homework Help
Replies
12
Views
2K

Hot Threads

Recent Insights

Back
Top