Explaining maths

Here is another one of my old posts, from way back in 2008. In fact I originally posted this here on Blogger, before I was persuaded to move over to WordPress. (In those days LaTeX wasn't available on Blogger. But now I can use MathJax here.) I deleted the version here on Blogger at some point, but now I have come full circle! I still think that the notion of absorption (and similar "quantifier packaging" ideas) could be very helpful to at least some students! But, as far as I know, the notion hasn't really really caught on, and I only have anecdotal evidence supporting my approach. See the WordPress post for comments on this idea from, among others, Professor Sir Tim Gowers. I'm repeating the post here mostly because the maths looks better here now using MathJax (though this particular example isn't a fair comparison, since the version of this post on WordPress doesn't actually use LaTeX). But also, it would be great if ...

Here is another question and (partial) answer from my FPM Piazza forum last autumn, this time related to the "challenge question" from my First quiz on permutations . First, here is a screenshot of the relevant quiz question. (You can click on the image to view it full size.) Challenge question from first FPM quiz on permutations

Here is another question and answer from my FPM Piazza forum last autumn. Question: Suppose that \(S\) is a set with two elements, say \(S=\{a,b\}\). When looking at elements of the Cartesian square \(S \times S\), are \((a,b)\) and \((b,a)\) the same element, or are they different elements? Does \(S \times S\) have four different elements, or only three? My answer: Hi, The key term in the definition of Cartesian squares, and generally Cartesian products, is "ordered pair". When you use standard round brackets in this way, the order does matter. You have specified a first coordinate and a second coordinate. For example, if you work in \(\mathbb{R} \times \mathbb{R} = \mathbb{R}^2\), the point \((1,0)\) (which lies on the x-axis) is different from the point \((0,1)\) (on the y-axis). Many of the sets \(S\) we have looked at are subsets of \(\mathbb{R}\), and this results in \(S \times S\) being a subset of \(\mathbb{R}^2\). When this happens, you can often th...

 My fourth FPM quiz last autumn was on sets and subsets. Again the "challenge" question isn't too hard, as long as you understand the basic concepts and definitions, though it is easy to make mistakes. Here is a screenshot of the question. As usual, the buttons don't do anything, but you can enlarge the image by clicking on it. Click on the screenshot to enlarge the image Here \(\mathbb{R}\), \(\mathbb{Q}\) and \(\mathbb{Z}\) have their usual meanings, and \(\emptyset\) is the empty set. You also need to know about the operations of intersection (denoted by \(\cap\)) and set difference (denoted by a backslash, \(\setminus\)) and the "subset" relation (denoted by \(\subseteq\)). Here I use the "subset or equals" notation \(\subseteq\) to make it clear that sets which are equal do count as subsets of each other. In particular, note that the notation \(Y \nsubseteq X\) means that \(Y\) is not a subset of \(X\), which is equivalent to saying tha...

My third FPM quiz last autumn was on prime factorization. As usual the final question was labelled as a "challenge question". This one is probably relatively easy once you really understand the definition of the set \(S\), which is closely related to the material from my classes on Bézout's lemma. (But you don't need to know about Bézout's lemma to answer the question.) Here is a screenshot of the question. As usual, the buttons don't do anything, but you can enlarge the image by clicking on it. Note that three of the statements are true and one of the statements is false. You are supposed to spot the false one!

 My second FPM quiz last autumn was a First quiz on rational and irrational numbers . Here is a screenshot of the challenge question. Note added: Thanks for asking about whether zero is a natural number. This varies in the literature, and (as you can check) the answer here depends crucially on this. For my teaching in Nottingham, zero is not included in the natural numbers. So the natural numbers are the strictly positive integers, i.e., \[\mathbb{N}=\{1,2,3,\dots\}\,.\]

Perhaps it would be interesting to post some of my so-called "challenge" questions from my FPM quizzes. I won't post them all at once. Here, as a png image, is the very first "challenge" question I set them, on GCDs (Greatest Common Divisors, also known as Highest Common Factors). It is just a screenshot, so the buttons don't do anything. Enjoy! Screenshot of "challenge" question from FPM First Quiz on GCDs