Let $A$ and $B$ be two sets such that $n\left(A\right)=3$ and $n\left(B\right)=2$.If $(x, 1),(y, 2),(z, 1)$ are in $A\times B$, find $A$ and $B$, where $x, y, z$ are distinct elements.

Discuss the following relation for reflexivity, symmetricity and transitivity:

The relation $R$ defined on the set of all positive integers by $\text{'}\text{'}mRn$ if $m$ divides $n\text{'}\text{'}$.

Discuss the following relation for reflexivity, symmetricity and transitivity:

Let $P$ denote the set of all straight lines in a plane. The relation $R$ defined by $\text{'}\text{'}lRm$ if $l$ is perpendicular to $m\text{'}\text{'}$

Discuss the following relation for reflexivity, symmetricity and transitivity:

Let $A$ be the set consisting of all the members of a family. The relation $R$ defined by $\text{'}\text{'}aRb$ if $a$ is not a sister of $b\text{'}\text{'}.$

Discuss the following relation for reflexivity, symmetricity and transitivity:

Let $A$ be the set consisting of all the female members of a family. The relation $R$ defined by $\text{'}\text{'}aRb$ if $a$ is not a sister of $b\text{'}\text{'}.$

Discuss the following relations for reflexivity, symmetricity and transitivity: 

On the set of natural numbers the relation $R$ defined by $\text{'}\text{'}xRy$ if $x+2y=1\text{'}\text{'}.$

Let $x=\left\{a,b,c,d\right\}$ and $R=\left\{\left(a,a\right),\left(b,b\right),\left(a,c\right)\right\}.$ Write down the minimum number of ordered pairs to be included to $R$ to make it Reflexive.

Let $x=\left\{a,b,c,d\right\}$ and $R=\left\{\left(a,a\right),\left(b,b\right),\left(a,c\right)\right\}.$ Write down the minimum number of ordered pairs to be included to $R$ to make it Symmetric