Let $A = \{1, 2, 3, 4\}$ and $R = \{(2, 2), (3, 3), (4, 4), (1, 2)\}$ be a relation on $A$. Then $R$ is:

Let $R$ be a relation from $Q$ to $Q$ defined by $R = \{(a, b) : a, b \in Q \text{ and } a - b \in Z\}$. Show that $(a, b) \in R$ implies that $(b, a) \in R$.

Determine whether the following relation $R$ in the set $A = \{1, 2, 3, 4, 5, 6\}$ defined by $R = \{(x, y) : y \text{ is divisible by } x\}$ is reflexive,symmetric,and transitive.

Let $R$ be the relation defined in the set $A = \{1, 2, 3, 4, 5, 6, 7\}$ by $R = \{(a, b) : \text{both } a \text{ and } b \text{ are either odd or even}\}$. Show that $R$ is an equivalence relation. Further,show that all the elements of the subset $\{1, 3, 5, 7\}$ are related to each other and all the elements of the subset $\{2, 4, 6\}$ are related to each other,but no element of the subset $\{1, 3, 5, 7\}$ is related to any element of the subset $\{2, 4, 6\}$.

The relation $S$ in the set $R$ of real numbers,defined as $S = \{(a, b) : a < b^2\}$ is a . . . . . . relation.

Let $A = \{-4, -3, -2, 0, 1, 3, 4\}$ and $R = \{(a, b) \in A \times A : b = |a| \text{ or } b^2 = a + 1\}$ be a relation on $A$. Then the minimum number of elements that must be added to the relation $R$ so that it becomes reflexive and symmetric is $........$.