Let $L$ be the set of all straight lines in the Euclidean plane. Two lines $l_1$ and $l_2$ are related by the relation $R$ if and only if $l_1$ is parallel to $l_2$. Then the relation $R$ is:

Let $L$ be the set of all lines in a plane and $R$ be the relation in $L$ defined as $R = \{(L_{1}, L_{2}) : L_{1} \text{ is perpendicular to } L_{2}\}$. Show that $R$ is symmetric but neither reflexive nor transitive.

Let $A = \{1, 2, 3\}$. The number of relations containing $(1, 2)$ and $(1, 3)$ which are reflexive and symmetric but not transitive is:

Let $A = \{1, 2, 3\}$. Show that the number of relations on $A$ containing $(1, 2)$ and $(2, 3)$ which are reflexive and transitive but not symmetric is $3$. (Note: The original prompt claimed $4$,but the correct count for this specific set is $3$).

Let $R$ be a relation on $\mathbb{R}$,given by $R = \{(a, b) : 3a - 3b + \sqrt{7} \text{ is an irrational number} \}$. Then $R$ is

Let $A = \{0, 3, 4, 6, 7, 8, 9, 10\}$ and $R$ be the relation defined on $A$ such that $R = \{(x, y) \in A \times A : x - y \text{ is an odd positive integer or } x - y = 2\}$. The minimum number of elements that must be added to the relation $R$ so that it becomes a symmetric relation is equal to $...........$.