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Category Archives: proof
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
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 bijection, finite, function, injection, invertible, one-to-one, onto, proof, surjection
7 Comments
A simple proof of the quadratic formula
If you’re reading this blog you have probably memorized (or used to have memorized) the quadratic formula, which can be used to solve quadratic equations of the form But do you know how to derive the formula? Usually the derivation … Continue reading
A combinatorial proof: PIE a la mode!
Continuing from my last post in this series, we’re trying to show that , where is defined as which is what we get when we start with a sequence of consecutive th powers and repeatedly take successive differences. Recall that … Continue reading
Posted in arithmetic, combinatorics, proof
Tagged consecutive, difference, function, integers, matching, powers
Comments Off on A combinatorial proof: PIE a la mode!
A combinatorial proof: counting bad functions
In a previous post we derived the following expression: . We are trying to show that , in order to show that starting with a sequence of consecutive th powers and repeatedly taking successive differences will always result in . … Continue reading
Posted in arithmetic, combinatorics, proof
Tagged consecutive, difference, function, integers, matching, powers
1 Comment
A combinatorial proof: functions and matchings
We’re trying to prove the following equality (see my previous post for a recap of the story so far): In particular we’re trying to show that the two sides of this equation correspond to two different ways to count the … Continue reading
Posted in arithmetic, combinatorics, proof
Tagged consecutive, difference, function, integers, matching, powers
5 Comments
A combinatorial proof: the story so far
In my last post I reintroduced this seemingly odd phenomenon: Start with consecutive integers and raise them all to the th power. Then repeatedly take pairwise differences (i.e. subtract the first from the second, and the second from the third, … Continue reading
Posted in arithmetic, combinatorics, proof
Tagged consecutive, difference, integers, powers
1 Comment
A combinatorial proof: reboot!
More than seven years ago I wrote about a curious phenomenon, which I found out about from Patrick Vennebush: if you start with a sequence of consecutive th powers, and repeatedly take pairwise differences, you always end up with , … Continue reading
Posted in arithmetic, combinatorics, proof
Tagged consecutive, difference, integers, powers
11 Comments
PIE: proof by counting
Recall the setup: we have a universal set and a collection of subsets , , , and so on, up to . PIE claims that we can compute the number of elements of that are in none of the (that … Continue reading
PIE: proof by algebra
In my previous post I stated a very formal, general form of the Principle of Inclusion-Exclusion, or PIE.1 In this post I am going to outline one proof of PIE. I’m not going to give a completely formal proof, because … Continue reading
The Math Less Traveled 