$A$ relation $R$ defined on a set $A$ is anti-symmetric if $(a, b) \in R$ and $(b, a) \in R$ implies:

Let $R$ be the relation on the set $\mathbb{R}$ of all real numbers defined by $a \ R \ b$ if $|a - b| \le 1$. Then $R$ is

Show that the number of equivalence relations on the set $A = \{1, 2, 3\}$ containing $(1, 2)$ and $(2, 1)$ is $2$.

Let $R$ and $S$ be equivalence relations on a set $A$. Then,

Let the relations $R_1$ and $R_2$ on the set $X = \{1, 2, 3, \ldots, 20\}$ be given by $R_1 = \{(x, y) : 2x - 3y = 2\}$ and $R_2 = \{(x, y) : -5x + 4y = 0\}$. If $M$ and $N$ are the minimum number of elements required to be added to $R_1$ and $R_2$,respectively,to make the relations symmetric,then $M + N$ equals

Show that the relation $R$ in the set of real numbers $\mathbb{R}$ defined as $R = \{(a, b) : a \leq b\}$ is reflexive and transitive but not symmetric.