Let $R = \{(L_1, L_2): L_1 \perp L_2\$ where $L_1, L_2 \in L$ (set of straight line in a plane)}, then

The relation $R = \{(L_1, L_2): L_1 \perp L_2\}$ where $L_1, L_2 \in L$ (set of straight lines in a plane) needs to be checked for reflexive, symmetric, and transitive properties.

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Checking if $R$ is reflexive:

A line cannot be perpendicular to itself. For two lines to be perpendicular, they must intersect at 90°, but a line and itself overlap completely.

Therefore $(L, L) \notin R$ for any line $L$.

$R$ is not reflexive.

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Checking if $R$ is symmetric:

If line $L_1$ is perpendicular to line $L_2$, then $L_2$ is also perpendicular to $L_1$. The angle between them is 90° regardless of which line is considered first.

If $(L_1, L_2) \in R$, then $(L_2, L_1) \in R$.

$R$ is symmetric.

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Checking if $R$ is transitive:

Consider:

• $L_2$ is a vertical line
• $L_1$ is a horizontal line, so $L_1 \perp L_2$
• $L_3$ is another horizontal line, so $L_2 \perp L_3$

However, $L_1$ and $L_3$ are both horizontal lines, making them parallel, not perpendicular.

Therefore, $L_1 \perp L_2$ and $L_2 \perp L_3$ does not imply $L_1 \perp L_3$.

$R$ is not transitive.

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The relation $R$ is symmetric but neither reflexive nor transitive.