Equivalence Relations

Example #4: Consider the set of rational numbers $\mathbb{Q}$ that are written as decimals, e.g., $0.5$, $1$, $1.25$, $-15.67$, etc.

Definition. [Integer Part] The integer part of a decimal number of the form $m.n$, where $m \in \mathbb{Z}$ and $n \in \mathbb{N}$, is the integer $m$. Notation: the integer part of a positive $k \in \mathbb{Q}$ is denoted by $\lfloor k \rfloor$ (

$\lfloor k \rfloor$

) and called the floor of $k$.

Examples: $\lfloor 0.5 \rfloor = 0$, $\lfloor 1 \rfloor = 1$, and $\lfloor 1.25 \rfloor = 1$. However, $\lfloor -15.67 \rfloor = -16$.

The relation that relates every two rational numbers whose integer parts are the same is an equivalence relation:

$\text{Same-Int-Parts} \;$$= \{(0, 0.5), \,$$(0, 0), \,$$(0.25, 0.5), \,$$\dots, \,$$(1, 1), \,$$(1, 1.25), \,$$\dots\}$.

Definition. [Ceiling] The operation of rounding up a real number $x \in \mathbb{R}$ is called taking the ceiling of $x$, or ceil shortly. Notation: the ceil of $x$ is denoted by $\lceil x \rceil$ (

$\lceil x \rceil$

).

Examples: $\lceil 0.5 \rceil = 1$, $\lceil 1 \rceil = 1$, $\lceil 1.25 \rceil = 2$, and $\lceil -15.67 \rceil = -15$ (so it is ceil that finds the integer part of negative nums!)