Set

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Introduction

The notion of a set is one of the fundamental notions in mathematics that has to be left undefined. Of course, we have plenty of synonyms for the word set like collection, ensemble, group, etc. but those names really do not define the meaning of the word set: all they can do is to replace it in various sentences. So, instead of defining what sets are, one has to define what can be done with them or, in other words, what axioms the sets satisfy. These axioms are chosen to agree with our intuitive concept of a set on one hand and to allow various, sometimes quite sophisticated, mathematical constructions on the other hand. For the full collection of these axioms, see Zermelo-Fraenkel Axioms. In this article we shall present just a brief discussion of the most common properties of sets and operations related to them.

Relation of belonging

The most important property of sets is that, for every object $x$ and a set $S$ , we can say whether $x$ belongs to $S$ (written as $x\in S$ ) or not (written as $x\notin S$ ). Two sets $S'$ and $S''$ are equal if they include the same objects, i.e., if for every object $x$ , we have $x\in S'$ if and only if $x\in S''$ .

Specifying a set by listing its objects

It means that in order to determine the set, it suffices to tell which objects belong to this set. If you have just several such objects, all you need to do is to list them. So, you can specify the set consisting of the numbers $1,3,5$ , and $239$ , for example. (The standard notation for this set is $\{1,3,5,239\}$ . note that the order in which the terms are listed is completely unimportant: we have to follow some order when writing things in one line but you should actually imagine those numbers flowing freely inside those curly braces with no preference given to any of them. What matters is that these four numbers are in the set and everything else is out). But how do you specify sets that have very many (maybe infinitely many) elements? You cannon list them all even if you spend your entire life writing!

Specifying a set by the common property of its elements

Another way to specify a set is to use some property to tell when an object belongs to this set. For instance, we may try to think (alas, only try!) of the set of all objects with green hair. In this case we do not even try to list all such objects. We just decide that something belongs to this set if it has green hair and doesn't belong to it otherwise. This is a wonderful way to describe a set. Unfortunately, it just doesn't work that simply. Indeed, consider the property that an object is a set that does not belong to itself (remember, that, given a set, we should be able to tell about every object whether it belongs to this set or not; in particular, we can ask this question about the set itself). It is easy to see that the set $S$ specified by this property can neither belong, nor not belong to itself and our whole theory gets self-contradictory. So, this way to describe sets should be used with extreme caution. The way out is that such a description is allowed only in much weaker form: given a set $S$ and a property $P$ , we can define a new set $S'$ of objects in the set $S$ that satisfy the property $P$ .

Subsets

We say that a set $A$ is a subset of a set $S$ if every object $x$ that belongs to $A$ also belongs to $S$ . For example, the sets $\{1,2\}$ and $\{2,3\}$ are subsets of the set $\{1,2,3\}$ but the set $\{1,6\}$ is not.