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Let L be the set of all straight lines i...

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 not

A

reflexive

B

symmetric

C

transitive

D

equivalence

Text Solution

Verified by Experts

The correct Answer is:
A, B, C, D
Relation R on the set of all straight lines in the plane is of parallel line.
A line is parallel to itself. So, R is reflexive.
If `l_(1)` is parallel to `l_(2)`, then `l_(2)` is parallel to `l_(1)`.
`therefore` R is symmetric relation. `[l_(1), l_(2) in L]`
Let `l_(1), l_(2), l_(3) in L`
`l_(1)` is parallel to `l_(2)` and `l_(2)` is parallel to `l_(3)`.
Then, `l_(1)` is parallel to `l_(3)`
`therefore R` is transitive relation.
So, R is equivalence relation.
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Knowledge Check

  • Relation " parallel" in the set of all straight lines in a plane is :

    A
    only reflexive
    B
    only symmetric
    C
    only transitive
    D
    equivalence relation

  • Let L be the set of all straight lines in plane. l_(1) and l_(2) are two lines in the set. R_(1), R_(2) and R_(3) are defined relations. (i) l_(1)R_(1)l_(2) : l_(1) is parallel to l_(2) (ii) l_(1)R_(2)l_(2) : l_(1) is perpendicular to l_(2) (iii) l_(1) R_(3)l_(2) : l_(1) intersects l_(2) Then which of the following is true ?

    A
    `R_(1), R_(2)` and `R_(3)` are equivalence
    B
    `R_(1)` is equivalence
    C
    `R_(2)` and `R_(3)` are reflexive
    D
    `R_(1), R_(2)` and `R_(3)` are not symmetric

  • Let R be the relation over the set of all straight lines in a plane such that l_(1) R l_(2) iff l_(1) _|_ l_(2) . Then, R is

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