Classical Sets

There are some special sets that are often used to group specific types of numbers. It is easy to write out a set containing even numbers from 1 to 10, $\{2,4,6,8,10\}$, but it is more difficult to write out sets that have infinitely many elements. Fortunately, there are special symbols to denote infinite sets of numbers such as the set of all integers, $\mathbb Z$, or the set of all rational numbers, $\mathbb Q$. These sets are used in many theoretical math problems.

Here is a list of common classical sets that include both countably infinite and uncountably infinite sets of numbers.

• $\mathbb C$ - The set of all complex numbers. This is an uncountably infinite set. $\mathbb C = \{a+bi \text{ such that } a,b \in \mathbb R\}$.

• $\mathbb R$ - The set of all real numbers. This is an uncountably infinite set.

• $\mathbb R_0$ - The set of all non-negative real numbers. This is an uncountably infinite set.

• $\mathbb R^+$ - The set of all positive real numbers. This is an uncountably infinite set.

• $\mathbb Q$ - The set of all rational numbers. This is a countably infinite set. $\mathbb Q = \{\frac{a}{b} \text{ such that } a,b \in \mathbb Z, \text{and }b \ne 0\}$.

• $\mathbb Q_0$ - The set of all non-negative rational numbers. This is a countably infinite set.

• $\mathbb Q^+$ - The set of all positive rational numbers. This is a countably infinite set.

• $\mathbb I$ - The set of all irrational numbers. This is an uncountably infinite set.

• $\mathbb Z$ - The set of all integers. This is a countably infinite set. $\mathbb Z = \{\dots, -2,-1,0,1,2,\dots\}$.

• $\mathbb Z_0$ - The set of all non-negative integers. This is a countably infinite set. $\mathbb Z_0 = \{0,1,2, \dots\}$.

• $\mathbb Z^+$ or $\mathbb N$ - The set of all positive integers. This is a countably infinite set. $\mathbb N = \{0,1,2,3, \dots \}$. Sometimes zero is excluded.

• $\mathbb P$ - The set of all prime numbers. This is a countably infinite set. $\mathbb P = \{2, 3, 5, 7, 11, 13, 17, \dots\}$

• $\mathbb Z_p$ - The set of integers modulo $p$. This is a finite set, $\mathbb Z_p = \{0,1,2, \dots\ p-1\}$ .

Some of the sets above are subsets of other sets listed.

Explain why $\mathbb P\subset\mathbb Z^+\subset\mathbb Z_0\subset\mathbb Z \subset Q$ is true.

Solution: $\mathbb P$ is the set of all prime numbers. All prime numbers are positive integers, but not all positive integers are prime numbers. Therefore, since $\mathbb Z^+$ is the set of all positive integers, $\mathbb P\subset\mathbb Z^+$. Since all positive integers are non-negative integers, $\mathbb Z^+\subset\mathbb Z_0$. This implies that $\mathbb P\subset\mathbb Z^+\subset\mathbb Z_0$. Since the set of all integers includes all non-negative integers, $\mathbb Z_0\subset\mathbb Z$, which implies $\mathbb P\subset\mathbb Z^+\subset\mathbb Z_0\subset\mathbb Z$. Finally, all integers are rational numbers, but not all rational numbers are integers, so $\mathbb Z \subset Q$. This implies $\mathbb P\subset\mathbb Z^+\subset\mathbb Z_0\subset\mathbb Z \subset Q$.

Some other relationships that can be derived in this manner are:

$\mathbb Q^+\subset\mathbb Q_0\subset\mathbb Q$

$\mathbb R^+\subset\mathbb R_0\subset\mathbb R\subset\mathbb C$

$\mathbb Q \cup \mathbb I = \mathbb R$