Question

Prove that the relation on $\mathbf{\mathbb{Z}}$defined by $\mathit{a}\mathbf{~}\mathit{b}$if and only if ${\mathbf{a}}^{\mathbf{2}}\mathbf{\equiv }{\mathbf{b}}^{\mathbf{2}}\mathbf{\left(}\mathbf{mod}\mathbf{6}\mathbf{\right)}$is an equivalence relation.

Short answer

We proved that $~$is an equivalence relation.

Step-by-step solution

To mention given data

Let the relation $~$be defined on $\mathrm{\mathbb{Z}}$by,

$a~b$if and only if $a^2\equiv b^2\left(mod6\right)$.

We have to prove that $~$is an equivalence relation.

In order to prove that$~$is an equivalence relation, we need to prove the following conditions:

1. '$~$’ is reflexive.
2. '$~$’ is symmetric.
3. '$~$’ is transitive.

To prove that ~ is reflexive

Let $a\in \mathrm{\mathbb{Z}}.$

Then:

localid="1660566769164" role="math" $$\begin{gathered} a~a \\ \iff a^2\equiv a^2\left(mod6\right) \end{gathered}$$

Hence, $~$is reflexive.

To prove that ~ is symmetric

Let$a,b\in \mathrm{\mathbb{Z}}$.

Then suppose that:

$$\begin{gathered} a~b \\ \iff a^2\equiv b^2\left(mod6\right) \\ b~a \\ \iff b^2\equiv a^2\left(mod6\right) \end{gathered}$$

Hence,$~$is symmetric.

To prove that ~ is transitive

Let $a,b,c\in \mathrm{\mathbb{Z}}$.

Then suppose that:

localid="1660566787101" $$\begin{gathered} a~b,b~c \\ \iff a^2\equiv b^2\left(mod6\right),b^2\equiv c^2\left(mod6\right) \\ \iff a^2\equiv c^2\left(mod6\right) \\ \iff a~c \end{gathered}$$

Hence,$~$is transitive.

Hence, $~$is an equivalence relation since all the conditions are satisfied.