Let $A=\left\{a,b\right\}$. List all relations on $A$ and find their number.

Let $R$ be a relation from $N$ to $N$ defined by $R=\left\{\right(a,b): a,b\in N \mathrm{a}\mathrm{n}\mathrm{d} a=b^2\}$.

Is the following statement is true for $\left(a,a\right)\in R$ for all $a\in N$?

Let $R$ be a relation from $N$ to $N$ defined by $R=\left\{\right(a,b): a,b\in N \mathrm{a}\mathrm{n}\mathrm{d} a=b^2\}$.

Is the following statement is true $\left(a,b\right)\in R\Rightarrow \left(b,a\right)\in R$?

Let $R$ be a relation from $N$ to $N$ defined by $R=\left\{\right(a,b) :a,b\in N \mathrm{a}\mathrm{n}\mathrm{d} a=b^2\}$

Is the following statement is true$\left(a,b\right)\in R$ and $\left(b,c\right)\in R\Rightarrow \left(a,c\right)\in R$?

Let $A=\left\{1,2,3,\dots ,14\right\}$. Define a relation on a set $A$ by

$R=\left\{\left(x,y\right): 3x-y=0, \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e} x,y\in A\right\}$.

Depict this relationship using an arrow diagram. Write down its domain, co-domain and range.

Define a relation $R$ on the set $N$ of natural numbers by

$R=\left\{\right(x,y):y=x+5,x$ is a natural number less than $4,x,y\in N\}$.

Depict this relationship using $\left(i\right)$ roster form $\left(ii\right)$ an arrow diagram. Write down the domain and range of $R$.

$A=\left\{\mathrm{1,2},\mathrm{3,5}\right\}$ and $B=\left\{4,6,9\right\}$. Define a relation $R$ from $A$ to $B$ by

$R=\{\left(x,y\right):$ the difference between $x$ and $y$ is odd, $x\in A, y\in B\}$.

Write $R$ in Roster form.