Give an example of a relation which is an empty relation.

Give an example of a relation which is a universal relation.

Let $R$ be a relation on $X$, If $R$ is symmetric then $xRy\Rightarrow yRx$. If it is also transitive then $x\mathrm{R}y\text{ and }y\mathrm{R}x\Rightarrow x\mathrm{Rx}$. So whenever a relation is symmetric and transitive then it is also reflexive. What is wrong in this argument?

Suppose a box contains a set of $n$ balls $\left(n\ge 4\right)$ (denoted by $B$) of four different colours (may have different sizes), viz. red, blue, green and yellow. Show that a relation $R$ defined on $B$ as $R=\left\{\left(b_1,b_2\right):\mathrm{balls} b_1 \mathrm{and} b_2 \mathrm{have} \mathrm{the} \mathrm{same} \mathrm{colour}\right\}$ is an equivalence relation. How many equivalence classes can you find with respect to $R$?
[Note: On any set $X$ a relation $R=\{\left(x,y\right):x \mathrm{and} y$satisfy the same property $P\}$ is an equivalence relation. As far as the property $P$ is concerned, elements $x$ and $y$ are deemed equivalent. For different $P$ we get different equivalence relations on $X$]

Find the number of equivalence relations on $X=\left\{1,2,3\right\}$.

Let $R$ be the relation on the set $\mathrm{ℝ}$ of real numbers such that $aRb$ iff $a-b$ is an integer. Test whether $R$ is an equivalence relation. If so find the equivalence class of $1$ and $\frac{1}{2}$ with respect to this equivalence relation.