Question

Let R be a relation on set L of lines defined by L1 R L2 if L1 perpendicular to L2 ,then relation R is
(a) reflexive and symmetry
(b)simmetric and transitive
(c)equivalance relation
(d) simmetric​

Answer 1

The relation R is symmetric but neither reflexive nor transitive.

Step-by-step explanation:

We are given that R be a relation on set L of lines defined by L1 R L2 if L1 perpendicular to L2.

The relation R = {($L_1,L_2$) : $L_1$ is perpendicular to $L_2$}

Firstly, we will check that the relation R is reflexive or not;

If R is reflexive, then (L, L) must belong to R;

Since the line L cannot be perpendicular to itself, this means that;

(L, L) ∉ R

Hence, the relation R is not reflexive.

Now, we will check that the relation R is symmetric or not;

If $L_1$ is perpendicular to $L_2$, then $L_2$ must also be perpendicular to $L_1$.

This means that $(L_1, L_2)$  belongs to R, then $(L_2, L_1)$ must also belong to R.

Hence, the relation R is symmetric.

Now, we will check that the relation R is transitive or not;

If $L_1$ is perpendicular to $L_2$, and $L_2$ is perpendicular to $L_3$, then it is sure that $L_1$ is not perpendicular to $L_3$.

This means that $L_1$ is parallel to $L_3$.

This means that is $(L_1, L_2)$  belongs to R, and $(L_2, L_3)$ belongs to R, then $(L_1, L_3)$ doesn't belong to R

Hence, the relation R is not transitive.