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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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