Proving Cardinal Number of R = Cardinal Number of {x|0<x<1}

In summary, the statement "proving cardinal number of R = cardinal number of {x|0<x<1}" means that the number of elements in the set of real numbers is equal to the number of elements in the set of numbers between 0 and 1. This is significant because it demonstrates the concept of infinity and how there can be different sizes of infinity. It can be proven using the Cantor-Bernstein-Schroeder theorem, which states that if there exists an injection from set A to set B and an injection from set B to set A, then the cardinalities of A and B are equal. An example of an injection from R to {x|0<x<1} is the function f(x)
  • #1
typhoonss821
14
1
Hey guys,

How can I proove the cardinal number of R is equal to the cardinal number of {x|0<x<1}??


Thanks~~~
 
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  • #2
find a bijection from [0,1] to R. Think of the tangent-function...
 

What is the meaning of "proving cardinal number of R = cardinal number of {x|0

The statement means that the cardinality (number of elements) of the set of real numbers (R) is equal to the cardinality of the set of numbers between 0 and 1, not including 0 and 1 themselves.

What is the significance of proving this cardinality equality?

This equality is significant because it shows that the sets of real numbers and numbers between 0 and 1 have the same number of elements, even though one set is a subset of the other. It also demonstrates the concept of "infinity" and how there can be different sizes of infinity.

How can this cardinality equality be proven?

This equality can be proven using the Cantor-Bernstein-Schroeder theorem, which states that if there exists an injection (one-to-one mapping) from set A to set B and an injection from set B to set A, then the cardinalities of A and B are equal.

What is an example of an injection from R to {x|0

An example of an injection from R to {x|0

What are the implications of this cardinality equality in mathematics?

This cardinality equality has implications in many areas of mathematics, such as number theory, set theory, and analysis. It also has implications in computer science and the study of algorithms, as it demonstrates the concept of different levels of infinity and the importance of understanding and defining sets.

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