Answer to Question #125899 in Discrete Mathematics for Krystal

Question #125899
Which of the following sets have the same cardinality? Select all that apply.


LaTeX: \mathbb{N} N

[0,1]

LaTeX: \mathbb{R} R

(0,1)
1
Expert's answer
2020-07-13T19:17:52-0400

Cardinality of R\mathbb{R} is the same as cardinality of (0,1), because there is a bijection from first set to second: 11+ex\dfrac{1}{1+e^{-x}}


Cardinality of [0,1] is the same as cardinality of (0,1), because there is a bijection from first set to second:

From the first set cut a sequence of points 0,1/4,1/42,1/43,0,1/4, 1/4^2, 1/4^3, \dots and insert them to coordinates 1/4,1/42,1/43,1/44,1/4, 1/4^2, 1/4^3, 1/4^4, \dots respectively. It is a bijection from [0,1] to (0,1].

Then set cut a sequence of points 1,(11/4),(11/42),(11/43),1,(1-1/4), (1-1/4^2), (1-1/4^3), \dots and insert them to coordinates (11/4),(11/42),(11/43),(11/44)(1-1/4), (1-1/4^2), (1-1/4^3), (1-1/4^4) \dots respectively. It is a bijection from (0,1] to (0,1).


Cardinality of N\mathbb{N} is less then cardinality of R\mathbb{R}, because N\mathbb{N} is a countable set and R\mathbb{R} is a continuum. R\mathbb{R} has the same cardinality as 2N2^\mathbb{N}.


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