Question

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)

Answer 1

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

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 $0,1/4, 1/4^2, 1/4^3, \dots$ and insert them to coordinates $1/4, 1/4^2, 1/4^3, 1/4^4, \dots$ respectively. It is a bijection from [0,1] to (0,1].

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

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