$A$ relation $R$ on the set of natural numbers is defined as $\{(a, b) : |a - b| = 3\}$. Then $R$ is:

Let $A = \{a, b, c\}$. The number of equivalence relations on $A$ containing $(b, c)$ is:

Let $A = \{1, 2, 3, 4, 6\}$. Let $R$ be the relation on $A$ defined by $R = \{(a, b) : a, b \in A, b \text{ is exactly divisible by } a\}$. Find the range of $R$.

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 $\{2, 4, 6\}$.

Let $S$ be the set of all real numbers. Then on the set $S$,the relation $R$ defined as $R = \{ (a, b) : 1 + ab > 0 \}$ is

Let $R$ be a transitive relation on a set $A$ and $I$ be the identity relation on $A$. Then: