Sets - Subsets of set of real numbers

The set of Real Numbers $\mathbb{R}$ serves as the universal set for many mathematical operations. It is composed of rational and irrational numbers.

Major subsets of $\mathbb{R}$ include: Natural numbers $\mathbb{N} = \{1, 2, 3, \dots\}$, Integers $\mathbb{Z} = \{\dots, -2, -1, 0, 1, 2, \dots\}$, and Rational numbers $\mathbb{Q}$.

The relationship between these subsets is expressed as $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$.

The set of Irrational numbers $T$ is also a subset of $\mathbb{R}$, where $T = \{x : x \in \mathbb{R} \text{ and } x \notin \mathbb{Q}\}$. Note that $\mathbb{Q} \cap T = \emptyset$.

Intervals are subsets of $\mathbb{R}$ and are used to represent continuous segments of the real line. They can be open $(a, b)$, closed $[a, b]$, or semi-open/semi-closed $[a, b)$ and $(a, b]$.

An open interval $(a, b)$ does not include the endpoints $a$ and $b$, whereas a closed interval $[a, b]$ includes both $a$ and $b$.