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Definiamo prodotto cartesiano AxB di due insiemi A e B l'insieme di tutte le coppie ordinate che hanno come primo elemento un elemento di A e come secondo elemento un elemento di B

Dati gli insiemi

A = { 1, 2, 3, 4 }
B = { a, b, c }
Costruisco tutte le coppie considerando come primo elemento un elemento di A e come secondo elemento un elemento di B
AxB = { (1,a) (1,b) (1,c) (2,a) (2,b) (2,c) (3,a) (3,b) (3,c) (4,a) (4,b) (4,c) }

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Given two sets A and B, set A is an improper subset of B if all elements of set A also belong to B and all elements of B also belong to A.

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One speaks of a subset precisely if at least one element is not included in the whole, ie in the case of strict inclusion.

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A Venn diagram (also called the Euler-Venn diagram) is a diagram showing all the possible logical relationships between a finite collection of different sets. These diagrams depict the elements as points in the plane, and the sets as regions enclosed by closed curves.

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Tabular representation

The tabular representation is obtained by enumerating the objects within braces;
example:
I want to consider set A composed of the first four natural numbers
1 2 3 4
I can write
A = {1, 2, 3, 4}
It is a representation that is used if the set is composed of a fairly limited number of objects

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The empty set is a set that has no elements. To indicate an empty set, use the symbol of a circle with an oblique bar ø or with two curly braces without any elements inside them {}.

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Consider a set A. It is said that a set B is a subset of A if and only if any element of B is taken, it is also an element of A. In this case we write that B⊂A or also A⊃B , and read "A is a subset of B"

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A set is a collection of elements determined by a precise rule that allows us to establish whether an element belongs to the whole, or if it does not belong to it.