Let L be the set of all straight lines in the Euclidean plane. Two lines $\ell_1$ and $\ell_2$ are said to be related by the relation R if $\ell_1$ is parallel to $\ell_2$. Then prove that R is equivalence relation.

Question

Let $L$ be the set of all straight lines in the Euclidean plane. Two lines $\ell_1$ and $\ell_2$ are said to be related by the relation $\mathrm{R}$ if $\ell_1$ is parallel to $\ell_2$. Then prove that $\mathrm{R}$ is equivalence relation.

Answer 1

$\quad \ell_1 \| \ell_2 \Rightarrow \mathrm{R}$ is reflexive.

$\ell_1\left\|\ell_2 \Rightarrow \ell_2\right\| \ell_1 \therefore \mathrm{R}$ is symmetric.

$\ell_1\left\|\ell_2 \Rightarrow \ell_2\right\| \ell_3 \Rightarrow \ell_1 \| \ell_3 \therefore \mathrm{R}$ is transitive.