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)

S seema garhwal

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

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 $y = 2x + c$ .