Let $L$ be the set of all straight lines in a plane and the relation $R$ on $L$ is defined by $\alpha R \beta \Leftrightarrow \alpha \perp \beta$,where $\alpha, \beta \in L$. Then $R$ is:

Let $A = \{2, 4, 6, 8\}$. $A$ relation $R$ on $A$ is defined by $R = \{(2, 4), (4, 2), (4, 6), (6, 4)\}$. Then $R$ is:

Let $X = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$. Let $R_{1}$ be a relation in $X$ given by $R_{1} = \{(x, y) : x - y \text{ is divisible by } 3\}$ and $R_{2}$ be another relation on $X$ given by $R_{2} = \{(x, y) : \{x, y\} \subset \{1, 4, 7\} \text{ or } \{x, y\} \subset \{2, 5, 8\} \text{ or } \{x, y\} \subset \{3, 6, 9\}\}$. Show that $R_{1} = R_{2}$.

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 $...........$.

Let $R = \{(1,2), (2,3), (3,3)\}$ be a relation defined on the set $A = \{1, 2, 3, 4\}$. Then the minimum number of elements needed to be added to $R$ so that $R$ becomes an equivalence relation is:

If $R$ and $S$ are two non-empty relations on a set $A$,then which of the following statements is false?