Hw2 Problem 1

### Problem 1

Let $S$ be the set of real numbers, $S = \mathbb{R}$. If $a,b \in S$, define $a\sim b$ if $a-b$ is an integer. Show that $\sim$ is an equivalence relation on $S$. Describe the equivalence classes of $S$.

Solution

(Reflexive)

Suppose $a \in S$, then $a-a = 0$ and $0$ is an integer. So, $a\sim a$ and $\sim$ is reflexive.

(Symmetry)

Suppose $a\sim b$, then $a-b=c$, where $c \in \mathbb{Z}$. By the additive property we find

(1)
\begin{align} a-b=c\\ a=c+b\\ -c=b-a\\ -1(c)=b-a \end{align}

Where $-1(c)$ is still an integer based on the properties of integer multiplication. So, $\sim$ is symmetric.

(Transitive)

Suppose $a\sim b$ and $b\sim c$. Then $a-b=z_1$, and $b-c=z_2$ where $z_1, z_2 \in \mathbb {Z}$. Then, using additive properties of reals.

(2)
\begin{align} a-(z_2 +c)=z_1\\ a-c-z_2 = z_1\\ a-c = z_1 + z_2 \end{align}

Further, by the additive properties of integers $z_1 + z_2 = z_3$, where $z_3 \in \mathbb{Z}$. So, $a-c = z_3$. Hence $a\sim c$ and $\sim$ is transitive.

The equivalence classes of $S$ would be $\mathbb {Z}$.