Sets

What is it?

In Mathematics, a set is just a collection of objects, which are called elements or members. The elements of a set can be any types of objects, including sets itself, and don’t need to share the same type between other elements.

It’s normally denoted with capital letters such as $A$, $B$, $S$, and $T$, and their generics elements by the corresponding lowercase letter $a$, $b$, $s$, and $t$. To indicate that a object $x$ is part of a set $X$, the notation will be $x\in X$. meaning that $x$ belongs to $X$.

Some special set of numbers have special notations, such as:

• $\mathbb{N}=\text{the set of natural numbers}$
• $z=\text{the set of integers}$
• $\mathbb{Q}=\text{the set of rational numbers}$
• $\mathbb{R}=\text{the set of real numbers}$

Elements of a set can be shown using the following notations:

• For a small number of elements

  $A=\text{the set of natural numbers not exceeding 7}=\{1,2,3,4,5,6,7\}$

• For infinite elements

  $\mathbb{Z}=\{1,2,3,\dots \}$

One also might denote the properties of a set using set-builder notation:

$$\mathbb{N}=\{x\in \mathbb{Z}|x>0\}$$

Where the left hand represents the membership and the right hand represents the properties of a set. The vertical bar $|$ is read as “such that”.j

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Subsets

Subsets can also be declared, and are defined as:

The set A is a subset of B if and only if, every element of A is also in B.

Which can be denoted as:

$$A\subset B⟺x\in A\to x\in B$$