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

$(a, b) \in R$ and $(b, c) \in R$ implies that $(a, c) \in R$

Let $R$ be the relation on $Z$ defined by $R = \{ (a,b):a,b \in Z,a - b$ is an integer $\} $  Find the domain and range of $R .$

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

Find the range of $R$

Determine the domain and range of the relation $R$ defined by $R =\{(x, x+5): x \in\{0,1,2,3,4,5\}\}$

Let $R$ be a relation from $N$ to $N$ defined by $R =\left\{(a, b): a, b \in N \text { and } a=b^{2}\right\} .$ Are the following true?

$(a, b) \in R,$ implies $(b, a) \in R$

The Fig shows a relation between the sets $P$ and $Q$. Write this relation 

in set - bulider form,

What is its domain and range ?