ZURAB JANELIDZE'S BLOG

What is FAM? "FAM" is an acronym for an undergraduate course, "Foundations of Abstract Mathematics", offered by the Department of Mathematical Sciences of Stellenbosch University (FAM I and FAM II, respectively) since 2012/2011. The course consists of two year-long modules, FAM I (Mathematics 278) and FAM II (Mathematics 378), offered at the second and the third year, respectively. It is possible to enroll for only one of the two modules. Neither of the modules has any prerequisites, although note that admission to the third-year module is subject to approval by the Department of Mathematical Sciences. The course aims to let the students experience mathematical research, at the level corresponding to students' mathematical skills, and in this process, to uplift those skills. A bit of history When teaching a course in calculus at the University of Cape Town in 2008, Zurab Janelidze was approached by a student, Pieter du Toit, with a request to show him mathemati...

The Kindest Ending A novel by  Natia Kuparadze and Zurab Janelidze  Chapter 1. Small Wooden House The morning was pleasantly cold and fragrant. The slightly painted balcony door gently let the spring sunlight into the bedroom, which slowly crept towards the middle of the room. Just a little, and the glittering rays would gently touch and caress the sleeping Noah's face. Noah loved his morning naps, when half-awake he convinced himself he was dreaming of a colorful world of his own making. These were magical moments that became more and more beautiful the closer they got to being given up. The rays of the sun flashed on his face.  Noah shook a little, then remained motionless, until finally he stretched his hands and opened his eyes. For a while he looked at the space of the room, as if he had not returned from the world of dreams.  He got up.  What would today be like? The mind's eye went over the day’s possible developments. He sorted them in his mind, ...

If you love collecting shells from a beach, you surely have at your home shells collected from several different beaches.  Let's say you have shells from three different beaches. Let B1 be the set of shells collected from the first beach, B2 the set of shells collected from the second beach, and B3 the set of shells collected from the third beach. Altogether, your collecting activity can be recorded by the set A={B1, B2, B3}. However, when you bring those shells at home, you will most likely mix them all up in one heap of shells. So if for instance, B1={S1, S2, S3}, B2={S4, S5} and B3={S6, S7, S8, S9}, where the symbols S1, ...., S9 represent the collected shells, then your collection at home, after the shells have been placed all together, is given by the set C={S1, S2, ..., S9}. Now, more generally, the union of a set A is the set C of all elements of individual elements of A. We write the union of A as UA, thus C=UA. Exploration 5.1. Compute the following unions (i.e., write do...

 A set can be pictured as a tree.  We describe this by examples. Consider for instance the set {A, B, {A, C, D, E}}. It has three elements, A, B, and the set {A, C, D, E}, which in turn has four elements, A, C, D, and E. Let us call this set X, so X={A, B, {A, C, D, E}}. Note that by coincidence, X and one of its elements, namely, {A, C, D, E}, share a common element A. This is permissible and it does not mean that there are two copies of A -- the A that is an element of X is the same A that is an element of X. Of course, in notation, we are forced to write out the set with two copies of A, but the two are supposed to represent the same object A. Now, to represent the set X as a tree, we start X and branch it out in as many elements as X has. So there will be three branches going out of X. The first two branches terminate with labels A and B, respectively. The third branch branches out further into four branches, which terminate with labels A, C, D, and E. As a result, we get ...

A singleton  is a set having exactly one element. For instance, {R} is a singleton, where R is a rose. Now, we do want to think of a rose R as a different object to the singleton {R}. These objects even have different types. The first is a plant, and the second is a set. Here is an illustrative analogy: a rose is not the same thing as as a vase with a rose in it. The empty set {} is not a singleton, since it does not have any elements. Neither is a set having two or more different elements a singleton. We can turn any object B into a singleton, by considering the set having B as its unique element, i.e., the set {B}. What happens when B={}? In notation, we get {B}={{}}. Now, what is the set {{}}? Is it still the empty set? Let us see. The empty set does not have any elements. What about the set {{}}? Does it have any elements? Well, this set is a singleton (remember, it is equal to {B}, where B={}), so yes, it has exactly one element, that element being the empty set B={}. So, sinc...

Elements in a set do not have to have any similar features. For instance, we can have a set {A1, A2, M1, M2, M3} where A1, A2 are apples and M1, M2, M3 are mountains. When we have a set whose elements are of different types, we often like to sort these elements by grouping elements of the same type together. Thus, for instance, in the set {A1, A2, M1, M2, M3} we may wish to group the apples together into one set {A1, A2} and group the mountains together into another set {M1, M2, M3}. This process is called a  partitioning  of a set, and the groupings we get as a result of partitioning are called  classes . Note that the classes are themselves sets, such as the class {A1, A2} and the class {M1, M2, M3} for the partitioning just discussed. Exploration 2.1.  Come up with real-life examples of partitioning and in each case, describe what the classes are. Here is a picture for inspiration: In the example we discussed, can we partition {A1, A2, M1, M2, M3} into the classes...

 Look at this photo of eight apples. How many apples are there in the photo? Eight. How many photos are there? One. Several objects or entities seen as one thing is called a set . The objects that make up the set are called its elements . Unlike in a photo, the arrangement of these objects among each other, or any information about the elements of the set apart from knowing what these elements are, is not considered to be part of the information about the set. Thus, for instance, the set of the eight apples that we see in the photo would be the same as the set of the same apples shown in a different photo, where these apples have been rearranged. If A1, A2, ..., A8 are symbolic representations of the apples shown in the photo, then {A1, A2, A3, A4, A5, A6, A7, A8} is how we symbolically represent the set of those apples.  Exploration 1.1. According to your interpretation of the definition of a set given above, which of the following sets should be the same set as the set ...

A well-known metaphor: a house is not a mere collection of bricks. It is, rather, a collection of bricks that has been organized in a certain structure. Organizing information into structure seems to be something our brains are good at. What if this organization is the means by which our brain comprehends, stores, and transmits information? In other words, the structure of information is all there is. "Meaning" may simply be certain types of structure of information that our brain distinguishes from others. When this distinction occurs, the brain provides us with emotional impulses, which creates this sensation of "aha, that is quite deep". If something like this is true, then the brain ought to be programmed to recognize structure in a way that is synchronized with the structure of the universe surrounding us, since those "aha" moments led us to a point where we can make predictions about nature, communicate over a large distance, etc.  It is difficult to...