Good morning, ladies and gentlemen in the Asian Pacific region, and good whatever-time-it-is-in-your-heathen-countries to the rest of the world. Today, we are going to Do Science.
Science is a very valuable and useful thing that humans use everyday, mostly for torturing poor students with fragments of the truth that are impossible to understand but also for wonderful and noble pursuits like figuring out whether buying a lottery ticket is a good idea (of course it is!), and whether or not you should grasp a hot plate directly with wet hands (don't do it).
See how much we've learned already? Science, people! It's important! Therefore, in the interests of making the world a better place, today I am going to explain to you the concept of sets, which is a very interesting (no, really, it is! Please don't stop reading) theory that kind of explains everything else in Mathematics. So, once you've got this, you're pretty much sorted.
The first gentleman to think about sets was not Georg Cantor (I know how to spell 'George', m'kay, his parents couldn't), but he was the first man to show why they were important and formally propose set theory. He was a German with a fancy beard and he said that 'a set is any collection into a whole of definite and distinct objects of our intuition or thought.'
There were some problems with Cantor's set theory though (notably the Russell paradox, which we shall discuss later), and in recent years it has been called the 'naive set theory' (ouch!). The modern definiton of sets is 'a set is a collection of objects which are distinct and distinguishable'.
Now, forget about the big words and the fancy numbers and all of that. Let's just think about common, everyday things, which is really what math is about in the first place. We use sets everyday! For example, if someone asks you who your best friends are, you might say, "My best friends are Adrian, Rose and Geetanjali.'
That, ladies and gentlemen, is a set. Does it fulfill all the conditions of a set? Let's check: first, it's a collection (of people sad enough to be friends with you), second the objects are distinct or non-repeated (unless Rose and Geetanjali are the other personalities of Adrian who is a schizophrenic, in which case some people might argue that this is not a set, and also, you probably need some more friends), and finally, the objects are distinguishable, in other words, we are able to decide whether they belong to the set or not. Consider Natalie. Is she one of your best friends? Whether the answer is, "No, and the bitch had better watch her back', or 'yes, we are totes BFFz', it proves that you are able to decide whether the object, Natalie, belongs to your set of best friends or not. Therefore, this meets all the conditions of being a set.
Simple, right? So what do all these terms like interjection, bijection, inverted and onto-onto mean? Patience, friends, I shall tell you.
First, let's discuss the different types of sets.
1) The Null Set: If you asked an unpopular person who their friends are, they would have to say, while weeping copiously, that they have no friends. We can represent this like this, A={}, where 'A' is the name we give the set of Sad Loser's friends, and the '{}' brackets contain the elements of this set. This set is empty, therefore we call it a null set.
2) The Finite Set: My textbook very helpfully describes this as a set that has a finite number of elements. Thanks, textbook. I would never have guessed that on my own. Finite basically means a number you can count or imagine, like the number of days left before my CET exam which I really should be preparing for.
3) The Infinite Set: My textbook again brilliantly describes this as a set which is not a finite set. *slow sarcastic applause*. In other words, this is a set of elements which are countless in number. Like a set of the number of stars in the sky, or the stress acne on my face because of these freaking exams.
4) The Universal Set: The universal set is basically the set of EVERYTHING. Of course, the definition of everything varies depending on the problem you're considering. If you're considering the problem of Indian politicians, say, we can define the universal set of this problem as the set of all the politicians in India, and this is a large set, but not an infinite one (thank goodness). However, if we're considering the problem of natural numbers (you know what natural numbers are. One, two, three, four.. It's not rocket science), this is an infinite set for obvious reasons.
5) The Subset: Nothing to do with sandwiches, unfortunately. Using the example given above, we can define a subset as a set, the elements of which are all elements of another set. Confused? A subset of Indian politicians, say B, can be defined as B={Narendra Modi, Rahul Gandhi, Arvind Kejriwal}. Each of the elements of this set B, is an element of the set 'Indian Politicians', therefore B is a subset of the universal set of Indian politicians. Kapish?
And those are some of the basics of set theory! There, that wasn't so hard, was it? We'll move on to operations in the next class, until then, have a good Thursday!
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