The number of reflexive relations on a set $A$ of $n$ elements is equal to

Determine whether the following relation is reflexive,symmetric,and transitive:
Relation $R$ in the set $N$ of natural numbers defined as
$R = \{(x, y) : y = x + 5 \text{ and } x < 4\}$

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

Let $A$ be a set consisting of $10$ elements. The number of non-empty relations from $A$ to $A$ that are reflexive but not symmetric is

In the set of all $3 \times 3$ real matrices,a relation is defined as follows: $A$ matrix $A$ is related to a matrix $B$ if and only if there exists a non-singular $3 \times 3$ matrix $P$ such that $B = P^{-1} A P$. This relation is