# Cardinality if infinite sets

#### novice

Let $$\displaystyle A=\{1,2,3\}$$ and $$\displaystyle B =\{a,b,c\}$$. Since they are finite sets, it’s quite obvious that they have the same number of elements.

I have read the proof that the infinite sets $$\displaystyle |(0,1)| = |\mathbb{R}|$$.
We know that $$\displaystyle (0,1) \subset \mathbb{R}$$, and I know there is a bijective function between the two sets, but how does one explain that $$\displaystyle (0,1)$$ and $$\displaystyle \mathbb{R}$$ have the same number of elements?

#### Plato

MHF Helper
I have read the proof that the infinite sets $$\displaystyle |(0,1)| = |\mathbb{R}|$$.
We know that $$\displaystyle (0,1) \subset \mathbb{R}$$, and I know there is a bijective function between the two sets, but how does one explain that $$\displaystyle (0,1)$$ and $$\displaystyle \mathbb{R}$$ have the same number of elements?
What does it mean ‘to have the same number’?
Each time I pick a number from $$\displaystyle (0,1)$$ you can match it with a unique number from $$\displaystyle \mathbb{R}$$.
Likewise you pick any number from $$\displaystyle \mathbb{R}$$ then I can match it with a unique number from $$\displaystyle (0,1)$$.
And the matches are all different because of the bijection.
You and I have the ‘same number’ of elements.

• novice

#### novice

What does it mean ‘to have the same number’?
Each time I pick a number from $$\displaystyle (0,1)$$ you can match it with a unique number from $$\displaystyle \mathbb{R}$$.
Likewise you pick any number from $$\displaystyle \mathbb{R}$$ then I can match it with a unique number from $$\displaystyle (0,1)$$.
You and I have the ‘same number’ of elements.
Let $$\displaystyle f 0,1) \rightarrow \mathbb{R}$$. Since there is a bijective function $$\displaystyle f$$ from $$\displaystyle (0,1)$$ to $$\displaystyle \mathbb{R}$$.

Let us say $$\displaystyle f(0.25)=3$$ and $$\displaystyle f(0.3)=5$$. If I remove equal number of elements from both sets, say $$\displaystyle (0,1)-\{0.25\}$$ and $$\displaystyle R-\{5\}$$. Now $$\displaystyle 0.25 \notin (0,1)$$ and $$\displaystyle 3 \in \mathbb{R}$$ , and $$\displaystyle 0.3 \in (0,1)$$ and $$\displaystyle 5 \notin \mathbb{R}$$.

Have we lost the bijective function?

Can we say $$\displaystyle |(0,1)-\{0.25\}|\not=|\mathbb{R}-\{5\}|$$?

#### Plato

MHF Helper
Let $$\displaystyle f 0,1) \rightarrow \mathbb{R}$$. Since there is a bijective function $$\displaystyle f$$ from $$\displaystyle (0,1)$$ to $$\displaystyle \mathbb{R}$$.

Let us say $$\displaystyle f(0.25)=3$$ and $$\displaystyle f(0.3)=5$$. If I remove equal number of elements from both sets, say $$\displaystyle (0,1)-\{0.25\}$$ and $$\displaystyle R-\{5\}$$. Now $$\displaystyle 0.25 \notin (0,1)$$ and $$\displaystyle 3 \in \mathbb{R}$$ , and $$\displaystyle 0.3 \in (0,1)$$ and $$\displaystyle 5 \notin \mathbb{R}$$.

Have we lost the bijective function?

Can we say $$\displaystyle |(0,1)-\{0.25\}|\not=|\mathbb{R}-\{5\}|$$?
I have no idea what any of that says. Much less what your point is.
If $$\displaystyle A$$ is any finite subset of $$\displaystyle (0,1)$$ and $$\displaystyle B$$ is any finite subset of $$\displaystyle \mathbb{R}$$ then $$\displaystyle \left| {\left( {0,1} \right)\backslash A} \right| = \left| {\mathbb{R}\backslash B} \right|$$

• novice