Q. 144.2( 54 Votes )

Let L be the set of all lines in XY plane and R be the relation in L defined as

R = {(L1, L2): L1 is parallel to L2}. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4.

Answer :

It is given that the relation in L defined as

R = {(L1, L2): L1 is parallel to L2}


R is reflexive as any line L1 is parallel to itself


(L1, L2) ϵ R


Now, Let (L1, L2) ϵ R


L1 is parallel to L2.


L2 is parallel to L1.


(L2, L1) ϵ R


Therefore, R is symmetric.


Now, Let (L1, L2), (L2, L3) ϵ R


L1 is parallel to L2. Also, L2 is parallel to L3.


L1 is parallel to L3.


(L1, L3) ϵ R


Therefore, R is transitive.


Therefore, R is an equivalence relation.


The set of all lines related to the line y = 2x + 4 is the set of all lines that are parallel to the line


y = 2x + 4


Slope of line y = 2x + 4 is m = 2


We know that parallel lines have the same slopes.


The line parallel to the given line is of the form y = 2x + c where, c ϵ R.


Therefore, the set of all lines related to the given line by y = 2x + c, where c ϵ R.


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