Some basics

OK, this post is supposed to be a gentle introduction to some of the words or terminologies used in math. What is a ‘set’? A set is a collection of distinct objects, eg: {apple, orange}, {football, basketball} etc.. As we can see, a set is written with ‘{‘, ‘}’ brackets, the examples are read as ‘a set of (or consisting of) apple and orange’ and so on. Individual entities of a set are called as ‘elements’ simply because they make up the set that we are interested in. As we can see, we can’t do much just with a set, well it’s not true cause lots of interesting objects are almost always defined using a set, but anyway what we really care about is ‘relations’ and ‘mappings’.

A mapping also called as a function can be thought of as a machine that takes an element from a set as input and gives us (or outputs) another element from a different set (or the same set). From the definition of a mapping, it is clear that we expect only one output for every input. A relation is then just a mapping of input and out put. A relation has to satisfy three other properties which are not really important for our discussion, so we will not go over them here.

Few more words to know, aarghh! The input set is called as the ‘domain’ and the output set is called as the ‘range’. I should mention that people use different words for ‘range’, one such word is ‘codomain’ and other is ‘image’ but this is very rare. So whenever anyone uses one of these words, clarify with them what they really mean, why? Let’s look at an example to understand why it is ambiguous. Imagine that we wanna know the efficiency of a car by measuring the mileage when we drive at different speeds. As everyone knows (look it up otherwise), the efficiency varies with different speeds and there is a certain range of speeds at which the car is the most efficient, let’s call the mileage at this speed to be 20 miles per gallon and the worst performance was 10 miles per gallon. For someone who knew nothing about how a car works, the mileage of this car can be any positive number but we know that the car can never go beyond 20 miles per gallon. So the codomain or the range in this case will be all positive numbers whereas the image (or the range, this is why it’s ambiguous) will be just all numbers between 10 and 20. I prefer to use codomain for the positive numbers and range for numbers between 10 and 20 in the last example. But I might use them interchangeably, I apologize for that, but I think the meaning will be clear from the context (hopefully).

These definitions for someone looking at it for the first time may seem to be general, well in fact it is general. Mathematicians have developed tools to study about these when the domain and the codomain have some special properties i.e., (surprise!!) when the domain and the codomain are numbers. So we will for the most part assume that the domain and the codomain are numbers of some sorts. To be precise, our domain will be r^n and the range will be r^n, what the heck is this? r or R is short for ‘real’ meaning all the numbers that we know, eg: -2, square root of 2, 10, 500 etc.. There are other things as well like the set of integers denoted by Z, natural numbers N, rational numbers Q, irrational numbers I, complex numbers C etc.. Curious readers can wiki it and those who have heard of them before know that each of them are what they are. For now let’s assume that n is equal to 1 i.e. all our functions are from r^1 to r^1 (written also as r to r). What makes these sets so special? Well there are lots of reasons and in fact huge part of what mathematicians do is identifying these sets and study them. Rambling apart, these are nice sets because any two elements of these sets can be compared i.e., 5 is greater than 6 (5>6), -square root of 2 is less than +square root of 2 and so on.

One special property of R, Q (and C) is that there exists an element in R, Q (and C) respectively for any element (other than zero) such that the product of both them is 1 or the unit element. Too many quantifiers in one single line, not cool brah! What I meant was that if we pick a number from r, let’s say 14, then 1/14 times 14 is 1 and this fact is true for any nonzero number we pick, that’s it! This doesn’t work if our set is Z, because 14 multiplied with any other integer will never be equal to 1. These sets are called as fields (to be pedantic, they also should satisfy other conditions but those are obvious ones like addition, multiplication etc.). Vector spaces are sets that are defined ‘over’ these fields like Q, R, C. Intuitively, (finite dimensional) vector spaces are tuples of elements from these fields i.e. an element from our favorite 2 dimensional vector space or plane is (3,4) or in general (x,y) where we require x and y to be elements of a field. So a n – dimensional vector space over a field is just the set of all n – tuples. Yep, that’s a very big set but these are the most nicest sets that one can encounter in their mathematical life, why? We’ll see that next time and more (I promise linearity II isn’t far, hang in there!).


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