Question

Let$\mathbf{~}$be defined on the set$\mathbf{ℝ}\mathbf{\text{'}}\mathbf{ℝ}\mathbf{\text{'}}\mathbf{ℝ}\mathbf{\text{'}}$of nonzero real numbers by:$\mathbf{a}\mathbf{~}\mathbf{b}$if and only if$\frac{\mathbf{a}}{\mathbf{b}\mathbf{\in }\mathbf{\mathbb{Q}}}$.Prove that$\mathbf{~}$is an equivalence relation.

Short answer

We proved that$~$ is an equivalence relation.

Step-by-step solution

To mention given data

Let$~~~$be defined on set${\mathrm{ℝ}}^{*}$of nonzero real numbers by:

$a~b$if and only if$\frac{a}{b}\in \mathrm{\mathbb{Q}}$ .

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 $a\in {\mathrm{ℝ}}^{*}$.

Then$a~a\iff \frac{a}{a=1},\forall a\in {\mathrm{ℝ}}^{*},1\in Q$ .

Hence, $~$is reflexive.

To prove that ~~ is symmetric

Let .Take $a,b\in {\mathrm{ℝ}}^{*}Takea=1,b=2$.

Then suppose that

$$\begin{gathered} a~b \\ \iff \frac{a}{b={\displaystyle \frac{1}{2\in \mathrm{\mathbb{Q}}(\because a=1,b=2)}}} \\ \iff \frac{b}{a={\displaystyle \frac{2}{1}}}\in Q \end{gathered}$$

$\iff b~a$

Hence,$~$ is symmetric.

To prove that ~~ is transitive

Let$a,b,c\in {\mathrm{ℝ}}^{*}a,b,c\in {\mathrm{ℝ}}^{*}$ .

Then suppose that

role="math" localid="1659168367849" $$\begin{gathered} a~b,b~c \\ \iff \frac{a}{b\in \mathrm{\mathbb{Q}}},\frac{b}{c\in \mathrm{\mathbb{Q}}} \\ \iff \frac{{\displaystyle \frac{a}{b}}}{{\displaystyle \frac{b}{c}}}=\frac{ac}{b^2}\in \mathrm{\mathbb{Q}} \end{gathered}$$

$\iff a~c,\forall a,b,c\in {\mathrm{ℝ}}^{*}$

Hence,$~$ is transitive.

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