ROUGH DRAFT authorea.com/107754

# Sets and Operators

The Set is a Mathematical Entity undefinable precisely: we could say that is a collection of elements of various species. We could imagine $$A$$ as the set of English Alphabet letters,or the set of the people living on the Earth, or a collection of songs, etc... . We could represent sets in different ways:

• Describing the property of the elements in braces:

$$A=\left\lbrace \right.$$The set of the letters of the English alphabet$$\left\rbrace \right.$$

• Saying the elements one-to-one:

$$A=\left\lbrace a,b,c,d, \ldots \right\rbrace$$

• The graphic representation by using the Euler-Venn Diagram:

Now if e.g. a letter a belongs to the set $$A$$ we will write $a\in A \,$ while if a number e.g. $$1$$ does not belong to $$A$$ we will write $1\not\in A.$ The set that does not contain elements is called Empty Set and is represented by the symbol $$\varnothing.$$
If we consider two sets $$A$$ and $$B$$ and one of them, $$B$$, is included in the other, $$A$$, we say that $$B$$ is subset of $$A$$ and we represents it by writing $B \subseteq A.$ We define Power Set of $$A$$, $$\mathcal{P}(A)$$, the set of all subsets of $$A$$, including $$A$$ and $$\varnothing$$. E.g.:

$$A=\lbrace a,b \rbrace$$

$$\mathcal{P}(A)=\big\lbrace \varnothing,A,\lbrace a \rbrace, \lbrace b \rbrace \big\rbrace$$

We note that if $$A$$ has $$n$$ elements, $$\mathcal{P}(A)$$ has $$2^{n}$$ elements.
It’s possible to define some operators over the sets that are: Union, Intersection,Complement and Cartesian Product.

• The Union represented by the symbol $$\cup$$ is the operator that given two sets $$A$$ and $$B$$ returns the set $$A \cup B$$ that contains the elements of both $$A$$ and $$B$$.
E.g.:

$$A=\lbrace a,b\rbrace$$
$$B=\lbrace c,d \rbrace$$
$$A \cup B =\lbrace a,b,c,d \rbrace$$

• The Intersection represented by the symbol $$\cap$$ is the operator that given two sets $$A$$ and $$B$$ returns the set $$A \cap B$$ that contains the elements that are in common between $$A$$ and $$B$$.
E.g.:

$$\ \ \ \ \ A=\lbrace a,b\rbrace$$
$$\ \ \ \ \ \ \ \ \ \ \ B=\lbrace a,b,c,d \rbrace$$
$$A \cap B =\lbrace a,b \rbrace$$

• The Complement represented by the symbol $$\setminus$$ is the operator that given two sets $$A$$ and $$B$$ returns the set $$A \setminus B$$ that contains the elements of $$A$$ that don’t belong to $$B$$.
E.g.:

$$\ \ \ \ \ \ \ \ \ \ \ A=\lbrace a,b,c,d \rbrace$$
$$\ \ \ \ \ B=\lbrace a,b\rbrace$$
$$A \setminus B =\lbrace c,d \rbrace$$

• The Cartesian Product represented by the symbol $$\times$$ is the operator that given two sets $$A$$ and $$B$$ returns the set $$A \times B$$ that is the set of the ordered pairs obtainable taking the first element by $$A$$ and the second one by $$B$$.

$$A=\lbrace a,b\rbrace$$
$$B=\lbrace c,d \rbrace$$
$$A \times B =\lbrace (a,c),(a,d),(b,c),(b,d) \rbrace$$

Starting by the Cartesian Product we will define the Relations.