Equivalence Relations

Fun Class Activities:

1. Definition. [Fraction Part] The fraction part of a decimal number of the form $m.n$, where $m \in \mathbb{Z}$ and $n \in \mathbb{N}$, is the rational number $0.n$.
   
   Example #1: the fraction part of $-15.67$ is $0.67$.
   Example #2: the fraction part of $1 = 1.0$ is $0.0 = 0$.
   
   Let's call the relation that relates every two rational numbers whose fraction part are the same $\text{Same-Frac-Parts}$. Is this relation an equivalence relation? Explain.
2. Is the $\text{Sibling}$ relation (that we had in one of our polls) an equivalence relation? Explain.
3. Is the $\text{LessThanOrEqualTo}$ ($\le$) relation an equivalence relation? Explain.