MCQ_SINGLE
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:
NUMERICAL
Let $A=\{1,2,3, \ldots \ldots \ldots \ldots, 100\}$. Let $R$ be a relation on $\mathrm{A}$ defined by $(x, y) \in R$ if and only if $2 x=3 y$. Let $R_1$ be a symmetric relation on $A$ such that $R \subset R_1$ and the number of elements in $R_1$ is $\mathrm{n}$. Then, the minimum value of $\mathrm{n}$ is _________.
MCQ_SINGLE
Let $P(S)$ denote the power set of $S=${$1, 2, 3, …, 10$}. Define the relations $R_1$ and $R_2$ on $P(S)$ as $AR_1B$ if $(A \cap B^c) \cup (B \cap A^c) = \emptyset$ and $AR_2B$ if $A \cup B^c = B \cup A^c$, $\forall A, B \in P(S)$. Then :
NUMERICAL
For $n \geq 2$, let $S_n$ denote the set of all subsets of $\{1,2, \ldots, n\}$ with no two consecutive numbers. For example $\{1,3,5\} \in S_6$, but $\{1,2,4\} \notin S_6$. Then $n\left(S_5\right)$ is equal to ________