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
$2^{89}-1$
$2^{89}-2^{45}$
$2^{45}-1$
$2^{90}-2^{45}$
Let $N$ be the set of natural numbers and a relation $R$ on $N$ be defined by $R=\left\{(x, y) \in N \times N: x^{3}-3 x^{2} y-x y^{2}+3 y^{3}=0\right\} .$ Then the relation $R$ is:
$R$ is a relation from $\{11, 12, 13\}$ to $\{8, 10, 12\}$ defined by $y = x - 3$. Then ${R^{ - 1}}$ is
Let a relation $R$ be defined by $R = \{(4, 5); (1, 4); (4, 6); (7, 6); (3, 7)\}$ then ${R^{ - 1}}oR$ is
Show that the relation $R$ in the set $R$ of real numbers, defined as $R =\left\{(a, b): a \leq b^{2}\right\}$ is neither reflexive nor symmetric nor transitive.
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