The 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:

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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:

H={1,3,5,7,9}

or

H={n : n positive, odd number and n < 1O}

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

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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)