Question

Let A be the set of all lines in the plane. Define a relation R on A as follows. For every l1 and l2 in A, l1 R l2 ⇔ l1 is parallel to l2. (Assume that a line is parallel to itself). Which of the following is true for R?

A. R is reflexive.
B. R is symmetric.
C. R is transitive.
D. R is neither reflexive, symmetric, nor transitive.

Answer 1

Hence, the relation R is a reflexive, symmetric and transitive relation.

Given :

A be the set of all lines in the plane and R is a relation on set A.

`R=\{l_1,l_2\in A|l_1 \;\text{is parallel to}\; l_2\}`

To find :

Which type of relation R on set A.

Explanation :

A relation R on a set A is called reflexive relation if every
`a\in A` then
`(a,a)\in R`.

So, the relation R is a reflexive relation because a line always parallels to itself.

A relation R on a set A is called Symmetric relation if
`(a,b)\in R` then
`(b,a)\in R` for all
`a,b\in A`.

So, the relation R is a symmetric relation because if a line
`l_1` is parallel to the line
`l_2` the always the line
`l_2` is parallel to the line
`l_1`.

A relation R on a set A is called transitive relation if
`(a,b)\in R` and
`(b,c)\in R` then
`(a,c)\in R` for all
`a,b,c\in A`.

So, the relation R is a transitive relation because if a line
`l_1` s parallel to the line
`l_2` and the line
`l_2` is parallel to the line
`l_3` then the always line
`l_1` is parallel to the line
`l_3`.

Therefore the relation R is a reflexive, symmetric and transitive relation.