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Symmetric Relations

Let L be th...

Question

Let $L$ be the set of all straight lines in the Euclidean plane. Two lines $l_{1}$ and $l_{2}$ are said to be related by the relation $R$ iff $l_{1}$ is parallel to $l_{2}$ . Then the relation $R$ is

reflexive

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symmetric

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transitive

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equivalence

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Solution

The correct options are
A reflexive
B transitive
C equivalence
D symmetric
In $x y$ co-ordinate plane.
Consider $3$ lines $l_{1}$ $l_{2}$ and $l_{3}$
Therefore
$l ∥ l$ hence R is reflexive
If $l_{1} | | l_{2}$ then this implies that $l_{2} | | l_{1}$ hence R is symmetric and
if $l_{1} | | l_{2}$ and $l_{2} | | l_{3}$ then $l_{1} | | l_{3}$ R is transitive
Hence the relation is symmetric, reflexive and transitive. Therefore it is an equivalence relation.

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