# 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 $y = 2x + 4.$

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

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 .

$\Rightarrow$L1 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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