Let $n$ be a fixed positive integer. Define a relation $R$ on the set $Z$ of integers by, $aRb \Leftrightarrow n|a - b$|. Then $R$ is

Let $R= \{(3, 3) (5, 5), (9, 9), (12, 12), (5, 12), (3, 9), (3, 12), (3, 5)\}$ be a relation on the set $A= \{3, 5, 9, 12\}.$ Then, $R$ is

Let $X =\{1,2,3,4,5,6,7,8,9\} .$ Let $R _{1}$ be a relation in $X$ given by $R _{1}=\{(x, y): x-y$ is divisible by $3\}$ and $R _{2}$ be another relation on $X$ given by ${R_2} = \{ (x,y):\{ x,y\}  \subset \{ 1,4,7\} \} $ or $\{x, y\} \subset\{2,5,8\} $ or $\{x, y\} \subset\{3,6,9\}\} .$ Show that $R _{1}= R _{2}$.

Let $R$ be the relation defined in the set $A=\{1,2,3,4,5,6,7\}$ by $R =\{(a, b):$  both $a$ and $b$ 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 $R$ be a relation over the set $N × N$ and it is defined by $(a,\,b)R(c,\,d) \Rightarrow a + d = b + c.$ Then $R$ is

Show that the number of equivalence relation in the set $\{1,2,3\} $ containing $(1,2)$ and $(2,1)$ is two.