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

On the power set $P(A)$ of a set $A$,the relation "is a subset of" $(\subseteq)$ is:

Let $A = \{1, 2, 3, 4\}$ and $R = \{(1, 2), (2, 3), (1, 4)\}$ be a relation on $A$. Let $S$ be the smallest equivalence relation on $A$ such that $R \subset S$. If the number of elements in $S$ is $n$,then the value of $n$ is:

Let $N$ denote the set of all natural numbers. Define two binary relations on $N$ as $R_1 = \{(x,y) \in N \times N : 2x + y = 10\}$ and $R_2 = \{(x,y) \in N \times N : x + 2y = 10\}$. Then

Let $A = \{2, 4, 6, 8\}$. $A$ relation $R$ on $A$ is defined by $R = \{(2, 4), (4, 2), (4, 6), (6, 4)\}$. Then $R$ is:

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