A new way to view 1-to-1 correspondences
One thing I love about teaching is that, no matter how many times you teach a certain course, you can still uncover new ways to explain the same concepts. This is exactly what happened earlier this semester in my mathematical reasoning course, when I was trying to help my students understand one-to-one correspondences, by use of pictures that I call “arrow diagrams.” In this short article, I’m going to explain what a one-to-one correspondence is, and then show the old and new ways of drawing diagrams for certain one-to-one correspondences.
So, what is a one-to-one correspondence?
A one-to-one correspondence (also called a bijection) is a way of pairing up numbers belonging to two different sets. Mathematicians say that the sets have the same size if a one-to-one correspondence can be constructed between them. The key is that:
- No number is used more than once
- No number is left out
If both of these conditions are met, then congratulations, you have found a one-to-one correspondence between your two sets. Note in general, for two given sets, a one-to-one correspondence will not be unique.
In the context of the mathematical reasoning course I am teaching, I introduce this concept to help students wrap their heads around different notions of infinity. For instance, sets that are countably infinite are referred to as sets of size aleph naught. Any set that has a one-to-one correspondence to the natural numbers is a set…