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Our goal is to get a rough idea of why $|\mathbb{R}| = |(0,1)|.$

To show that the sets $\mathbb{R}$ and $(0,1)$ are the same size, we would need to find a one-to-one correspondence between the elements of $\mathbb{R}$ and the elements of $(0,1)$ .

We describe one-to-one correspondences using some assignment rule or pattern. When we did this with the sets $\mathbb{N}$ and $2\mathbb{N}$ in Basics of Set Theory, the pattern was: send every element of $\mathbb{N}$ to its double in $2\mathbb{N}$ . This was a concise way of describing the assignment rule.

Mathematicians call this type of assignment rule a function. In the earlier example, we could write the function between $\mathbb{N}$ and $2\mathbb{N}$ as $f(x) = 2x$ . In other words, if you input $x$ into the function, it outputs $2$ times $x$ .

Some, but not all, one-to-one correspondences can be written concisely as functions. Not every function between two sets is a one-to-one correspondence though! We still have to check that it sends exactly one element to exactly one element, and doesn't miss any.

In the case of $\mathbb{R}$ and $(0,1)$ , there is a nice function that we can use, and it is in fact a one-to-one correspondence. In the In-Depth explanation we'll go into more detail about why this function is a one-to-one correspondence between the set $(0,1)$ and $\mathbb{R}$ . In this section we’ll just tell you what the function is and show you a picture.

The function actually starts with an element of the set $(0,1)$ , and assigns an element of $\mathbb{R}$ . So let $x$ be an element of $(0,1)$ . Then the function assigns $x$ to $f(x) = \tan(\pi x - \pi/2)$ .

The phrase "tan" means this is a form of the tangent function, from trigonometry. (The reason we wanted to leave $t=0$ and $t=1$ out of our set corresponding to breathed infinity, is because this function $f$ breaks if we try to plug in $0$ or $1$ .) Here is a graph of this function.

Figure: Graph of shifted and compressed tangent function.

You can read a short explanation of why $|\mathbb{R}| = |(0,1)|$ , or an in-depth explanation (a rigorous proof), or you can take our word for it, and go read the story or the meta-discussion.

Short Version Meta Discussion In-Depth Version True Story

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