Types of Relations

The relation $R$can be defined in the set$\left\{1,2,3,4,5,6\right\}$ as  $R=\left\{(a,b):b=a+1\right\}$ .It is an example of 

A relation $R$ on the set $A=\left\{a,b,c\right\}$is said to be symmetric if 

A relation $R$ is said to be reflexive for all $a,b\in A$, if  

Let $R=\left\{\left(a,b\right):a \text{divides} b\right\}$ defined on set of natural numbers; then $R$ is

Let $R=\left\{\left(L_1,L_2\right):L_1 \text{is perpendicular to} L_2\right\}$ defined on the set of all lines in $XY-$plane, then $R$ is

$R=\left\{\left(x,y\right):x<y\right\}$ defined on $N$ is

Let $A=\left\{1,2,3\right\}$, and $R=\left\{\left(1,1\right),\left(2,1\right),\left(2,2\right),\left(1,2\right),\left(3,2\right),\left(3,3\right)\right\}$; then $R$ is:

Let $A=\left\{1,2,3\right\}$, and $R=\left\{\left(1,1\right),\left(1,2\right),\left(1,3\right),\left(2,1\right),\left(2,3\right)\right\}$; then $R$ is:

Let $A=\left\{1,2,3\right\}$, which of the following is not an equivalence relation on $A?$

Let $R$ be the relation in the set $\left\{1,2,3\right\};$ given by $R=\left\{\left(1,2\right),\left(2,2\right),\left(1,1\right),\left(2,3\right),\left(1,3\right),\left(3,3\right),\left(3,2\right)\right\}.$ Choose the correct answer.

Let$R=\left\{\left(1,3\right),\left(2,4\right),\left(4,2\right),\left(2,3\right),\left(3,1\right)\right\}$ be a relationship on set $A=\left\{1,2,3,4\right\}$. The relation is 

Which of the following is not an equivalence relation on $Z?$

A relation $R$ on the set $A=\left\{a,b,c\right\}$ is said to be transitive, if

Show that the relation $R$ in the set of real numbers defined as:

$R=\left\{\left(a,b\right):a⩽b^3\right\}$ is not an equivalence relation.

Show that the relation $R$ in the set of real numbers defined as:

$R=\left\{\left(a,b\right):a⩽b\right\}$ is reflexive, transitive but not symmetric.

Show that the relation $R$ in the set of real numbers defined as:

$R=\left\{\left(a,b\right):a^2+b^2=1\right\}$ is symmetric but neither reflexive nor transitive.

Let $A$ be the set of all the points in a plane and let $O$ be the origin. Show that the relation $R$ defined by $R=\left\{\left(P,Q\right):OP=OQ\right\}$ is an equivalence relation.

Let $A=\left\{1,2,3\right\}, R_1=\left\{\left(1,1\right),\left(1,2\right),\left(2,1\right),\left(2,2\right),\left(3,3\right)\right\}$ and $R_2=\left\{\left(1,1\right),\left(2,2\right),\left(3,3\right),\left(2,3\right),\left(3,2\right)\right\}$. Are $R_1$ and $R_2$ equivalence relations$?$ Is $R_1\cap R_2$ is an equivalence relation$?$ Give reason to support your answer.

Let $R$ be a relation on the set 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 relation $R$ is an equivalence relation.

Show that the relation $R$ in the set of all polygons as $R=\left\{\left(P_2,P_1\right): P_1 \text{and} P_2 \text{have same number of sides}\right\}$, is an equivalence relation. What is the set of all elements in $A$ related to the right angle triangle $T$ with sides $3,4 \text{and} 5?$