Question

Let A be the set of all lines in XY-plane and let R be the relation in A defined by
$R=\left\{ \left( {{L}_{1}},{{L}_{2}} \right):{{L}_{1}}\parallel {{L}_{2}} \right\}$.
Show that R is an equivalence relation in A.
Find the set of all lines related to the line y = 3x+5

Answer 1

Hint: Recall the definition of an equivalence relation. A relation is said to be an equivalence relation if it is reflexive, symmetric and transitive. Prove that the given relation is reflexive, symmetric as well as transitive. Observe that the lines which are mapped by the given relation will have the same slope. Hence find the set of all lines related to the line y=3x+5.

Complete step-by-step answer:
Before solving the question, we need to understand what is an equivalence relation.
Reflexive relation: A relation R on a set “A” is said to be reflexive if $\forall a\in A$ we have $aRa$.
Symmetric relation: A relation R on a set “A” is said to be symmetric if $aRb\Rightarrow bRa$
Transitive relation: A relation R on a set “A” is said to be transitive if $aRb,bRc\Rightarrow aRc$.
Equivalence relation: A relation R on a set “A” is said to be an equivalence relation if the relation is reflexive, symmetric and transitive.

Reflexivity: Since every line is parallel to itself, we have for every line l in XY-plane l is related to l. Hence the relation is reflexive.
Symmetricity: We have $lRm\Rightarrow l\parallel m\Rightarrow m\parallel l\Rightarrow mRl$. Hence the relation is symmetric.
Transitivity: We have $lRm,mRn\Rightarrow l\parallel m,m\parallel n$
Since two lines parallel to the same line are parallel to each other, we have
$l\parallel n\Rightarrow lRn$
Hence, the relation is transitive.
Since the relation R is reflexive, symmetric and transitive, R is an equivalence relation.
Two lines parallel to each other in the XY-plane will have the same slope. Hence any line related to the line y = 3x+5 will have the same slope as that if y = 3x+5.
Comparing y = 3x+5 with slope intercept form of the line y = mx+c, we get m = 3.
Hence the slope of the line y = 3x+5 is 3
Hence any line related to y = 3x+5 will have slope 3.
Hence any line related to y = 3x+5 will have the equation of the form y = 3x+c.
Hence the set of lines related to y = 3x+5 is given $\left\{ y=3x+c,c\in \mathbb{R} \right\}$

Note: [1] Students usually make a mistake while proving reflexivity of a relation. In reflexivity, we need all the elements of a to be related with themselves, and even if a single element is found such that it is not related with itself, then the relation is not reflexive.
[2] The set of lines related to y = 3x+5 is called an equivalence class of the relation.
In equivalence class, we need to know only one element to know the whole class. An equivalence class is defined only for an equivalence relation.