Reviewer in Math About Sets

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  • Topic: Set theory, Set, Subset
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Reviewer in Math

A set is a collection of well defined and distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics. Developed at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived. In mathematics education, elementary topics such as Venn diagrams are taught at a young age, while more advanced concepts are taught as part of a university degree. -------------------------------------------------

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Definition
A set is a well defined collection of objects. Georg Cantor, the founder of set theory, gave the following definition of a set at the beginning of his Beiträge zur Begründung der transfiniten Mengenlehre:[1] A set is a gathering together into a whole of definite, distinct objects of our perception [Anschauung] and of our thought – which are called elements of the set. The elements or members of a set can be anything: numbers, people, letters of the alphabet, other sets, and so on. Sets are conventionally denoted with capital letters. Sets A and B are equal if and only if they have precisely the same elements.[2] As discussed below, the definition given above turned out to be inadequate for formal mathematics; instead, the notion of a "set" is taken as an undefined primitive in axiomatic set theory, and its properties are defined by the Zermelo–Fraenkel axioms. The most basic properties are that a set "has" elements, and that two sets are equal (one and the same) if and only if every element of one is an element of the other. -------------------------------------------------

Introduction to Sets
Forget everything you know about numbers.
In fact, forget you even know what a number is.
This is where mathematics starts.
Instead of math with numbers, we will now think about math with "things". Definition
What is a set? Well, simply put, it's a collection.
First you specify a common property among "things" (this word will be defined later) and then you gather up all the "things" that have this common property. | For example, the items you wear: these would include shoes, socks, hat, shirt, pants, and so on.I'm sure you could come up with at least a hundred.This is known as a set.|

Or another example would be types of fingers.This set would include 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. You simply list each element, 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't really an 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:

* The first set {socks, shoes, watches, shirts, ...} we call an infinite set, * the second set {index, middle, ring, pinky} we call a finite set. 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 a finite 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,...
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