Basic concepts of set theory

The notion (concept) of a set is one of the initial notions of mathematics that can’t be defined quite formally. We can talk about a set of triangle edges (sides), about the set of Russian words, about the set of natural numbers and so on. Synonyms of the term “set” are: a collection, a group, a class etc. Approximately one may say that a set is a collection of objects or ideas called elements of the set.

From the examples above, we can see that the notion of a set may be applied to collections of different nature:

ü A set of triangle sides (It contains a small, finite number of elements).

ü A set of Russian words (It contains a very large number, because nobody knows how many Russian words there exist. But, for sure, this number is less than, say, one billion. So, this set is finite).

ü A set of natural numbers (It contains an infinite amount of elements, because this set can’t be listed even theoretically. But we can easily decide if a given number is a natural one or not).

Sets are usually denoted by Latin capital letters and their elements – by corresponding small letters.

It is important to notice that for any set A the following is true: A, A A.

Two sets A and B are equal (denoted by A=B) if they consist of the same elements. This is equivalent to the condition that, for arbitrary x, if it is contained in A then it follows that x is in set B as well, and vice versa. So:

For any xÎ A è (it follows) xÎ B and for any yÎ B è yÎ A then A = B.

The number of elements in a set A is called the cardinality of A and is denoted │ A│. If A = {a, b, c, d} then │ A│ = 4.

Set P (A) is a power set of A. P (A) contains all possible subsets of set A.
Ex.: A = {a, b}. P (A) = {Æ, {a}, {b}, {a, b}}. Elements of a power set are sets themselves.

There is a difference between “a” and “{a}”. {a} is not an element of A. It is a subset of A and it is an element of P (A).)

Example. Let the set of Ivory Coast population be {Alex, Bob, …, Sam, … }. Suppose Sam is the only member of Ivory Cost team for Winter Olympic games. In such a case the set of national teams at the games = {{US team}, {Russia team}, {Canada team}, …, {Sam}, …}. Now Sam is treated as a team. He has a team status, like the status of the US Olympic team of about 600 members.

Definition: sets A and B (finite or infinite) are called equivalent (A B) if they have the same cardinality.

Ex.1:

A = {1, 7, 10, 15}, B = {a, b, c, d}. │ A│ = │ B│ A B.

Ex.2:

C = {x│ x = , i N}, D = {x│ x = i -3, i N}.

│ C│ = │ N│, │ D│ = │ N│ │ C│ = │ D│ C D.

H/t: Prove that the number │ P (A) │ of all subsets of a finite set A equals to .