The number of relations, on the set $\{1,2,3\}$ containing $(1,2)$ and $(2,3)$, which are reflexive and transitive but not symmetric, is

Let $R$ be a relation defined on $N \times N$ by $(a, b) R(c, d) \Leftrightarrow  a(b + c) = c(a + d).$ Then $R$ is

Let $\mathrm{A}=\{1,2,3,4,5\}$. Let $\mathrm{R}$ be a relation on $\mathrm{A}$ defined by $x R y$ if and only if $4 x \leq 5 y$. Let $m$ be the number of elements in $\mathrm{R}$ and $\mathrm{n}$ be the minimum number of elements from $\mathrm{A} \times \mathrm{A}$ that are required to be added to $\mathrm{R}$ to make it a symmetric relation. Then $m+n$ is equal to:

Let $R_{1}$ and $R_{2}$ be two relations defined on $R$ by $a R _{1} b \Leftrightarrow a b \geq 0$ and $a R_{2} b \Leftrightarrow a \geq b$, then

Consider the following two binary relations on the set $A= \{a, b, c\}$ : $R_1 = \{(c, a) (b, b) , (a, c), (c,c), (b, c), (a, a)\}$ and $R_2 = \{(a, b), (b, a), (c, c), (c,a), (a, a), (b, b), (a, c)\}.$ Then

Give an example of a relation. Which is Transitive but neither reflexive nor symmetric.