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# Category Archives: logic

## 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

## The wizard’s rational puzzle (solutions, part 2)

At long last, here is the solution I had in mind for the Wizard’s rational puzzle. Recall that the goal is to figure out the numerator and denominator of a secret rational number, if all we are allowed to do … Continue reading

Posted in arithmetic, challenges, logic, programming, puzzles, solutions
Tagged arithmetic, binary, denominator, Euclidean, gcd, logarithm, numerator, puzzle, rational, search, wizard

## The wizard’s rational puzzle (solutions, part 1)

About two and a half months ago I posted a challenge involving a sadistic math wizard, metal cubes containing rational numbers, and a room full of strange machines. I’ve been remiss in following up with some solutions. (Go read the … Continue reading

Posted in arithmetic, challenges, logic, programming, puzzles, solutions
Tagged arithmetic, denominator, Euclidean, gcd, logarithm, numerator, puzzle, rational, wizard
3 Comments

## The wizard’s rational puzzle (mind your p’s and q’s!)

You have been abducted by a sadistic math wizard (don’t you hate it when that happens?). He ushers you into a plain but cozy-looking room, with a hardwood floor, a few exotic-looking rugs, and wood paneling on the walls. He … Continue reading

Posted in arithmetic, challenges, logic, programming, puzzles
Tagged arithmetic, denominator, numerator, puzzle, rational, wizard
15 Comments

## Book review: Roads to Infinity

What is infinity? What is proof? These are two of the biggest questions mathematicians have grappled with over the years. In this well-written and fascinating book, John Stillwell takes us on a tour through some of the answers to these … Continue reading

Posted in arithmetic, books, computation, induction, infinity, logic, proof, review
Tagged infinity, John Stillwell, proof, roads

## Manufactoria

A friend of mine just pointed me to a most excellent puzzle game, Manufactoria, wherein you build little machines to test robots. For now I won’t give away the secret of what real math/computer science topic the game teaches you, … Continue reading

## The haybaler

At Penn Alexander’s math club yesterday, the students worked on a fun puzzle that I’d never seen before. It goes like this: You have five bales of hay. For some reason, instead of being weighed individually, they were weighed in … Continue reading