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Author Archives: Brent
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
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
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
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 computer, game, Lean, natural, number, proof
6 Comments
An exploration of forward differences for bored elementary school students
Last week I made a mathematics worksheet for my 8yearold 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 coronavirus, difference, elementary, forward, homeschool, worksheet
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 onetoone and onto. So prove that is onetoone, and prove that … Continue reading
Posted in logic, proof
Tagged bijection, finite, function, injection, invertible, onetoone, onto, proof, surjection
7 Comments
Onesided 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 bijection, function, injection, invertible, onetoone, onto, surjection
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