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

Answer:

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.