Relations and its Types

Let $R=\left\{\right(x,y);x+2y<6, x,y\in N\}$. Find the range of $R$.

Check the relation on the set $\mathrm{A}=\left\{1,2,3,4,5,6\right\}$ by $\mathrm{R}=(\mathrm{a},\mathrm{b}) \in \mathrm{R}:|\mathrm{a}-\mathrm{b}|\ge 0$ is the universal relation.

 Let A be the set of all students of a boys school. Show that the relation $\mathrm{R}$ on $\mathrm{B}$ given by $\mathrm{R}=\mathrm{\{}(\mathrm{a},\mathrm{b}):$ difference between the heights of $\mathrm{a}$ and$\mathrm{b}$ is less than $4$ meters} is the universal-relation.

Define universal relation on sets.

Verify the following relation is an identity relation or not:

If $\mathrm{A}=\left\{\mathrm{p},\mathrm{q},\mathrm{r},\mathrm{s}\right\}$ and ${\mathrm{I}}_{\mathrm{A}}=\left\{\left(\mathrm{p},\mathrm{p}\right), \left(\mathrm{q},\mathrm{q}\right), \left(\mathrm{r},\mathrm{r}\right), \left(\mathrm{s},\mathrm{s}\right)\right\}$.

If $\mathrm{A}=\left\{7, 8, 9\right\}$ and $\mathrm{R}$ is the relation on set $\mathrm{A}$, then find the antisymmetric relation on set $\mathrm{A}$.

In the set $\mathrm{N}$ of natural numbers, a relation $\mathrm{R}$ is defined by $\mathrm{aRb}\iff \mathrm{a}$ is a factor of $\mathrm{b}$. Examine if $\mathrm{R}$ is Anti-symmetric.

If $\mathrm{A}=\left\{\mathrm{p}, \mathrm{q}, \mathrm{r}, \mathrm{s}\right\}$ and $\mathrm{R}$ is the relation on set $\mathrm{A}$, then find the antisymmetric relation on set $\mathrm{A}$.

If $\mathrm{A}=\left\{2, 3, 4, 5\right\}$ and $\mathrm{R}$ is the relation on set $\mathrm{A}$, then find the antisymmetric relation on set $\mathrm{A}$.

The diagram shows the graph of a relation $R$ that maps set $A$ onto set $B$.

Write down $R$ as a set of ordered pairs.

Express the mapping $R$ in the form $x\to ...$

Consider the relation $\left\{\right(1,1),(1,2),(2,1\left)\right\}$. Classify it as one-to-one, many-to-one, many-to-many or one-to-many.

Consider the relation $\left\{\right(1,1),(1,2),(1,3\left)\right\}$. Classify it as one-to-one, many-to-one, many-to-many or one-to-many.

Consider the relation $\left\{\right(1,1),(2,1),(3,1\left)\right\}$. Classify it as one-to-one, many-to-one, many-to-many or one-to-many.

Consider the relation $\left\{\right(1,2),(2,3),(3,6\left)\right\}$. Classify it as one-to-one, many-to-one, many-to-many or one-to-many.

Consider the relation $\left\{\right(1,2),(2,3),(1,4\left)\right\}$. Classify it as one-to-one, many-to-one, many-to-many or one-to-many.

Is $g=\left\{\right(1,1),(2,3),(3,5),(4,7\left)\right\}$ a function? If $g$ is described by $g\left(x\right)=\alpha x+\beta$, then what value should be assigned to $\alpha$ and $\beta$.

Let $A=\{a, b, c\}$ and the relation $R$ be defined on $A$ as follows:

$R=\left\{\right(a,a),(b,c),(a,b\left)\right\}$.

Then, write minimum number of ordered pairs to be added in $R$ to make $R$ reflexive and transitive.

The relation $R$ in the set $A=\{1,2,3,4,5\}$ given by $R=\left\{\right(a,b):|a-b|$ is even}, is

Let $R$ be a relation on the set $A$ of ordered pairs of positive integers defined by $\left(x, y\right) R \left(u, v\right)$ if and only if $xv=yu.$ Show that $R$ is an equivalence relation.