Author Archives: Brent

About Brent

Associate Professor of Computer Science at Hendrix College. Functional programmer, mathematician, teacher, pianist, follower of Jesus.

A few words about PWW #33: subset permutations

My previous post showed four rows of diagrams, where the th row (counting from zero) has diagrams with dots. The diagrams in the th row depict all possible paths that start at the top left dot, end at the top … Continue reading

Posted in combinatorics, posts without words | Tagged , , , | 6 Comments

Post without words #33

Posted in combinatorics, posts without words | Tagged , , | 4 Comments

Happy tau day!

Happy Tau Day! The Tau Manifesto by Michael Hartl is now available in print, if you’re in the market for a particularly nerdy coffee table book and conversation starter: I also wrote about Tau Day ten years ago; see that … Continue reading

Posted in famous numbers, links | Tagged , , | 1 Comment

Challenge: area of a parallelogram

And now for something completely different!1 Suppose we have a parallelogram with one corner at the origin, and two adjacent corners at coordinates and . What is the area of the parallelogram? There are probably many different ways to derive … Continue reading

Posted in challenges, geometry | Tagged , | 18 Comments

Post without words #32

Posted in posts without words | Tagged , , , | 8 Comments

The Natural Number Game

Hello everyone! It has been quite a while since I have written anything here—my last post was in March 2020, and since then I have been overwhelmed dealing with online and hybrid teaching, plus a newborn (who is now almost … Continue reading

Posted in challenges, computation, proof | Tagged , , , , , | 6 Comments

An exploration of forward differences for bored elementary school students

Last week I made a mathematics worksheet for my 8-year-old son, whose school is closed due to the coronavirus pandemic. I’m republishing it here so others can use it for similar purposes. Figurate numbers and forward differences There are lots … Continue reading

Posted in arithmetic, teaching | Tagged , , , , , | 1 Comment

Ways to prove a bijection

You have a function and want to prove it is a bijection. What can you do? By the book A bijection is defined as a function which is both one-to-one and onto. So prove that is one-to-one, and prove that … Continue reading

Posted in logic, proof | Tagged , , , , , , , , | 7 Comments

One-sided inverses, surjections, and injections

Several commenters correctly answered the question from my previous post: if we have a function and such that for every , then is not necessarily invertible. Here are a few counterexamples: Commenter Buddha Buck came up with probably the simplest … Continue reading

Posted in logic | Tagged , , , , , , | 1 Comment

Test your intuition: bijections

Suppose we have sets and and a function (that is, ’s domain is and its codomain is ). Suppose there is another function such that for every . Is necessarily a bijection? That is, does necessarily match up each element … Continue reading

Posted in logic | Tagged , , , | 12 Comments