Two Worlds of Relations
Terminology and definitions in mathematics sometimes vary according to context. Here we’ll look at the ideas of relations, functions, and their domains, and discover that they look different from different perspectives.
Definitions
Cartesian Product of Sets
(A new question of the week) I had a long discussion recently about the Cartesian product of sets, answering questions like, “How is it Cartesian?” and “How is it a product?” I like discussions about the relationships between different concepts, and people who ask these little-but-big questions. We’ll be looking at about a quarter of …
What is a Set … Really?
Sometimes the more basic an idea is, the harder it is to define it. It is also very hard to understand a definition in English when you are not a native speaker! We have had some interesting discussions of such issues recently with a student who asks very basic and yet very challenging questions of …
Proving Certain Polynomials Form a Group
Abstract algebra can be a huge leap for many students, who may know algebra well, but are not used to abstraction – generalizing the concept of numbers so we can invent new kinds of “numbers” and “operations” and comparing their properties. Here we will look at a question from a student beginning the study of …
Sines Without Right Triangles
(A new question of the week) A recent question dealt with an apparent conflict between the right-triangle definition of sines and cosines, and the unit-circle definition, pertaining to multiples of 90° (angles on the axes). This provides an opportunity to look closely at the relationship between those two definitions. Two definitions Recall that the right-triangle …
Is {0} Closed Under Division? Thoughts, and Second Thoughts
(A new question of the week) A set is closed under an operation if, whenever that operation is applied to two elements of the set, the result is still an element of the set. It’s straightforward … until you look closely at some details! In the course of the discussion, we’ll dig into different definitions …
Is {0} Closed Under Division? Thoughts, and Second Thoughts Read More »
Average Rate of Change of a Function
(A new question of the week) Average rate of change is a topic taught in pre-calculus and calculus courses, primarily as preparation for the derivative, though it has more immediate applications. A recent question asked about when the concept is valid, which I found interesting.
Four Kinds of “Mean”
Last week, we looked at exactly what the mean is, referring specifically to the arithmetic mean, the one we first learn as the “average”. But just as we previously saw that there are several things called “average” (mean, median, mode), there are in fact several different kinds of “mean”. We’ll look here at the arithmetic, …
Three Kinds of “Average”
There are three different statistics that are commonly taught as “averages”, or “measures of central tendency”, of a set of numbers: mean, median, and mode. (There are others as well, which we will get to later.) What are they? How do they differ? How do you use them? We’ll look into questions like these as …
The Symmetric Derivative
To close out this series on the definition of the derivative, I want to look at a few questions about alternative versions of the definition, primarily the “symmetric difference quotient”. We’ll see that this leads to a slightly different result, not always equivalent to the original, and we’ll observe some associated ways that calculators can …