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Let L be the set of all lines in XY plane and R be the relation in L defined as R = {(L 1 , L 2 ) : L 1 is parallel to L 2 }. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4.

Q.14   Let L be the set of all lines in XY plane and R be the relation in L defined as
R = \{(L_1 , L_2 ) : L_1\;is\;parallel\;to\;L_2 \}. Show that R is an equivalence relation. Find
the set of all lines related to the line y = 2x + 4.

 

Answers (1)
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R = \{(L_1 , L_2 ) : L_1\;is\;parallel\;to\;L_2 \}

All lines are parallel to itself, so it is reflexive.

Let,

(L_1,L_2) \in R  i.e.L1 is parallel to T2.

L1 is parallel to L2 is same as L2 is parallel to L1 i.e. (L_2,L_1) \in R

Hence,it is symmetric.

Let,

(L_1,L_2) \in R  and  (L_2,L_3) \in R  i.e. L1 is parallel to L2  and L2 is parallel  to L3 .

\RightarrowL1 is parallel to L3   i.e. (L_1,L_3) \in R

Hence, it is transitive,

Thus,  R = \{(L_1 , L_2 ) : L_1\;is\;parallel\;to\;L_2 \} , is equivalence relation.

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

Here,  Slope = m = 2      and     constant = c = 4

 It is known that slope of parallel lines are equal.

Lines parallel to this ( y = 2x + 4. )  line  are y = 2x + c  , c \in R

Hence, set of all parallel lines to y = 2x + 4. are y = 2x + c.

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