User talk:Majesty awharitefe

GENERAL MATHEMATICS

set
Definition What is a set? Well, simply put, it'sa collection. First we specify a common property among "things" (this word will be defined later) and then we gather up all the "things" that have this common property. For example, the items you wear: shoes, socks, hat, shirt, pants, and so on. I'm sure you could come up with at least a hundred. This is known as aset. Or another example istypes of fingers. This set includes index, middle, ring, and pinky. So it is just things grouped together with a certain property in common. Notation There is a fairly simple notation for sets. We simply list each element (or "member") separated by a comma, and then put some curly brackets around the whole thing: The curly brackets{ }are sometimes called "set brackets" or "braces". This is the notation for the two previous examples: {socks, shoes, watches, shirts, ...} {index, middle, ring, pinky} Notice how the first example has the "..." (three dots together). The three dots...are called an ellipsis, and mean "continue on". So that means the first example continues on ... for infinity. (OK, there isn'treallyan infinite amount of things you could wear, but I'm not entirely sure about that! After an hour of thinking of different things, I'm still not sure. So let's just say it is infinite for this example.) So: But sometimes the "..." can be used in the middle to save writing long lists: Example: the set of letters: {a, b, c, ..., x, y, z} In this case it is afinite set(there are only 26 letters, right?) Numerical Sets So what does this have to do with mathematics? When we define a set, all we have to specify is a common characteristic. Who says we can't do so with numbers? Set of even numbers: {..., -4, -2, 0, 2, 4, ...} Set of odd numbers: {..., -3, -1, 1, 3, ...} Set of prime numbers: {2, 3, 5, 7, 11, 13, 17, ...} Positive multiples of 3 that are less than 10: {3, 6, 9} And the list goes on. We can come up with all different types of sets. There can also be sets of numbers that have no common property, they are justdefinedthat way. For example: {2, 3, 6, 828, 3839, 8827} {4, 5, 6, 10, 21} {2, 949, 48282, 42882959, 119484203} Are all sets that I just randomly banged on my keyboard to produce. Why are Sets Important? Sets are the fundamental property of mathematics. Now as a word of warning, sets, by themselves, seem pretty pointless. But it's only when we apply sets in different situations do they become the powerful building block of mathematics that they are. Math can get amazingly complicated quite fast. Graph Theory, Abstract Algebra, Real Analysis, Complex Analysis, Linear Algebra, Number Theory, and the list goes on. But there is one thing that all of these share in common:Sets. Universal Set At the start we used the word "things" in quotes. We call this theuniversal set. It's a set that contains everything. Well, notexactlyeverything.Everything that is relevant to our question. Then our sets included integers. The universal set for that would be all the integers. In fact, when doing Number Theory, this is almost always what the universal set is, as Number Theory is simply the study of integers. However in Calculus(also known as real analysis), the universal set is almost always the real numbers. And in complex analysis, you guessed it, the universal set is the complex numbers. Some More Notation When talking about sets, it is fairly standard to use Capital Letters to represent the set, and lowercase letters to represent an element in that set. So for example, A is a set, and a is an element in A. Same with B and b, and C and c. Now you don't have to listen to the standard, you can use something likemto represent a set without breaking any mathematical laws (watch out, you can getπyears in math jail for dividing by 0), but this notation is pretty nice and easy to follow, so why not? Also, when we say an elementais in a setA, we use the symbolto show it. And if something is not in a set use. Example: SetAis {1,2,3}. We can see that1A, but5A Equality Two sets are equal if they have precisely the same members. Now, at first glance they may not seem equal, so we may have to examine them closely! Example: Are A and B equal where: Let's check. They both contain 1. They both contain 2. And 3, And 4. And we have checked every element of both sets, so:Yes, they are equal! And the equals sign (=) is used to show equality, so we write: A = B Subsets When we define a set, if we take pieces of that set, we can form what is called asubset. So for example, we have the set {1, 2, 3, 4, 5}. Asubsetof this is {1, 2, 3}. Another subset is {3, 4} or even another, {1}. However, {1, 6} is not a subset, since it contains an element (6) which is not in the parent set. In general:
 * .The first set{socks, shoes, watches, shirts, ...}we call aninfinite set,
 * .the second set{index, middle, ring, pinky}we call afinite set.
 * .A is the set whose members are the first four positive whole numbers
 * .B = {4, 2, 1, 3}