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# Author Archives: Brent

## 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 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 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

## 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 bijection, function, injection, invertible, one-to-one, 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

## Book review: Beautiful Symmetry

[Disclosure of Material Connection: MIT Press kindly provided me with a free review copy of this book. I was not required to write a positive review. The opinions expressed are my own.] Beautiful Symmetry: A Coloring Book about Math Alex … Continue reading

## Hypercube offsets

In my previous posts, each drawing consisted of two offset copies of the previous drawing. For example, here are the drawings for and : You can see how the drawing contains an exact copy of the drawing, plus another copy … Continue reading

## Post without words #31

Posted in posts without words
Tagged binary, bits, cube, diagram, Hasse, hypercube, lattice, subsets
2 Comments

## A few words about PWW #30

A few things about the images in my previous post that you may or may not have noticed: As several commenters figured out, the th diagram (starting with ) is showing every possible subset a set of items. Two subsets … Continue reading

Posted in posts without words
Tagged binary, bits, cube, diagram, Hasse, hypercube, lattice, subsets
4 Comments

## Post without words #30

Posted in posts without words
Tagged binary, bits, cube, diagram, Hasse, hypercube, lattice, subsets
5 Comments

## The First Six Books of the Elements of Euclid, by Oliver Byrne (Taschen)

Recently for my birthday I received a copy of Oliver Byrne’s 1847 edition of Euclid’s Elements (pictured at right), republished by Taschen Books in 2010. I’ve only just started reading it, but it’s beautiful and fascinating. Oliver Byrne was a … Continue reading