Question

If m and n are lines in a plane P, define$\mathit{m}\mathbf{~}\mathit{n}$ to mean that m and n are parallel. Is $\mathbf{~}$an equivalence relation on P?

Short answer

It is proved that given relation$~$ is equivalence.

Step-by-step solution

Definitions of Equivalence relation

A relation R defined on a set is called an equivalence relation if it is reflexive, symmetric and transitive.

Prove that ~is an equivalence relation on P

Objective is to prove that $~$is an equivalence relation. That is to show that the relation $~$is reflexive, symmetric and transitive.

For reflexive,

Let$m~m$ as each line is parallel to itself. So,$~$ is reflexive.

For symmetric,

Let$m~n$ that is$m\parallel n$ which is same as$n\parallel m$ that is$n~m$ . Thus,$m~n$ implies$n~m$. So,$~$ is symmetric.

For transitive,

Let$m~n$, $n~l$then $m\parallel n$and$n\parallel l$. Clearly,$m\parallel l$ that is.$m~l$ Thus$m~n$,$n~l$, implies$m~l$ . So,$~$ is transitive.

Therefore, the relation$~$ is an equivalence relation.