The minimum number of elements that must be added to the relation $R =\{( a , b ),( b , c )$, (b, d) $\}$ on the set $\{a, b, c, d\}$ so that it is an equivalence relation, is $.........$

If $R_{1}$ and $R_{2}$ are equivalence relations in a set $A$, show that $R_{1} \cap R_{2}$ is also an equivalence relation.

Show that the relation $R$ defined in the set $A$ of all polygons as $R=\left\{\left(P_{1}, P_{2}\right):\right.$ $P _{1}$ 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 ${R_1}$ be a relation defined by ${R_1} = \{ (a,\,b)|a \ge b,\,a,\,b \in R\} $. Then ${R_1}$ is

Show that the relation $R$ in the set $\{1,2,3\}$ given by $R =\{(1,2),(2,1)\}$ is symmetric but neither reflexive nor transitive.

Define a relation $R$ over a class of $n \times n$ real matrices $A$ and $B$ as $"ARB$ iff there exists a non-singular matrix $P$ such that $PAP ^{-1}= B "$ Then which of the following is true?