Institute of Information Science - University of Miskolc

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Mathematical //set// is the totality of well-defined objects. The objects which belong to the set are called the //elements//.

This state of belonging to a given set is called relation, which has the following sign: a∈X

You may read the given relation like this: “the elements of X set”, or “X set contains the following elements”.

The elements of a set:

{{:tanszek:oktatas:techcomm:information_-_basics:pasted:20231026-204435.png}}

The number of the elements which belong to a give set is called the cardinality of the set. This cardinal may be finite or infinite. 

The elements (which belong to a given set) can be defined by enumeration or by giving an exact principle of how they belong to that set.

**For example**:

The set of natural numbers:

$$ Z = \{1, 2, 3, 4, 5, ...\} $$ 

You may define the elements by writing them: List the natural, odd integers from one to ten:

$$ A = \{1, 3, 5, 7, 9\} $$

or

$$  A = \{x \mid x \text{ is an odd integer and } 1 \leq x < 10 \} $$

or in modern C++:
<sxh c++>
std::set<int> mySet = {1, 2, 3};
</sxh>

or in Python:
<sxh Python>
my_set = {1, 2, 3}
</sxh>


The sets (and their belongings) are usually set in a sharp way.

{{:tanszek:oktatas:techcomm:information_-_basics:pasted:20231026-204536.png}}

In information technology it is possible to use sets which contain fuzzy elements as well.

In these cases the value of how an element is connected to a given set is defined by a '//membership function//' 
μm(x)

{{:tanszek:oktatas:techcomm:information_-_basics:pasted:20240930-173642.png}}

//Fuzzy sets// are a generalization of classical sets used in mathematics and logic to //handle uncertainty// and //partial membership//. Unlike classical sets where an element either belongs or does not belong to a set (membership is binary: 0 or 1), fuzzy sets allow for degrees of membership, represented by values between 0 and 1.

Further reading: [[https://towardsdatascience.com/a-very-brief-introduction-to-fuzzy-logic-and-fuzzy-systems-d68d14b3a3b8|A very brief introduction to fuzzy-logic]]