The concept of a set is intuitive and it could be defined as a "collection of objects". Thus, we can talk of a set of people, cities, glasses, pens or of the set of objects on a table in a given moment. A set is well defined once one can know if a given element may belong to it or not. The set of blue pens is well defined, because we can tell if a pen is blue by looking at it. The set of tall people is not well defined, because we are not always able to tell if a person is tall by just looking or also different people may have different opinions about who is tall. In the nineteenth century, according to Frege, the elements of a set were only defined by a property.
A set is a group, class or collection of objects referred to as elements of the set (although any definition implicitly hides logical paradoxes or contradictions). By object we mean not only physical entities, such as tables, chairs, etc., but also abstract entities such as numbers, letters, etc. The membership relation between elements and sets is always perfectly discernible; in other words, whether an object belongs to a set is always regarded as true or false.
A set can be determined in two ways: by extension or by compression.
Determination of a set by extension
A set is determined by extension when all its elements are written one by one.
The numbers less than $$5$$: $$A=\{1,2,3,4\}$$.
Determination by compression
A set is determined by compression when only a feature common to all the elements is mentioned.
The set of vowels of the alphabet: $$X=\{x: \ x \text{ is a vowel}\}$$.
Notation
We will call element each of the objects that are part of a set. These elements have an individual character, qualities that allow us to differentiate them, and each one is unique. There are no duplicates or repeated elements. They will be denoted with a lower case letter: $$a, b, c,\ldots$$
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$$\in$$ / $$\notin$$: Used to express whether an element belongs to a set or not.
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$$\subset$$: Used to express that a set, and therefore all its elements, are part of a larger set.
- $$ U $$ / $$\emptyset$$: The first symbol denotes the universal set, which is the set of all the things we are dealing with. Thus, if we talk about integers then U is the set of integers; if we talk about cities, U is the set of all cities. This universal set can be mentioned explicitly, or, as in most of the cases, can be assumed, given the context in which we are working. However, it is always necessary to prove the existence of such a set previously. The other set is called the empty set and it satisfies that all possible elements are not contained in it, that is $$\forall x, \ x\notin\emptyset$$.