EXERCISE 1.1

Give an example of a relation. Which is reflexive and transitive but not symmetric.

Give an example of a relation. Which is symmetric and transitive but not reflexive.

Show that the relation $R$ in the set $A$ of points in a plane given by $R=\left\{\right(P,Q):$ distance of the point $P$ from the origin is same as the distance of the point $Q$ from the origin $\}$, is an equivalence relation. Further, show that the set of all points related to a point $P\ne \left(0,0\right)$ is the circle passing through $P$ with origin as center.

Show that the relation $R$ defined in the set $A$ of all triangles as $R=\{\left(T_1,T_2\right):T_1$ is similar to $T_2\},$ is equivalence relation. Consider three right angle triangles $T_1$ with sides $3,4,5, T_2$ with sides $5,12,13$ and $T_3$ with sides $6,8,10.$ Which triangles among $T_1,T_2$ and $T_3$ are related?

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

Let $L$ be the set of all lines in $XY$ plane and $R$ be the relation in $L$ defined as $R=\left\{\right(L_1,L_2):L_1$ is parallel to $L_2\}.$ Show that $R$ is an equivalence relation. Find the set of all lines related to the line $y=2x+4.$

Let $R$ be the relation in the set $\{1,2,3,4\}$ given by $R=\left\{\right(1,2),(2,2),(1,1),(4,4),(1,3),(3,3),(3,2\left)\right\}$. Choose the correct answer.

Let $R$ be the relation in the set $N$ given by $R=\left\{\left(a,b\right) : a=b –2, b>6\right\}.$