Question

Let $S=\{1,2,3,4,5,6\}$. Then the number of one-one functions $f: S \rightarrow P ( S )$, where $P ( S )$ denote the power set of $S$, such that $f(m) \subset f(m)$ where $n < m$ is _______

Answer 1

Let $S=\{1,2,3,4,5,6\}$, then the number of one-one functions, $f : S \cdot P ( S )$, where $P ( S )$ denotes the power set of $S$, such that $f(n)< f(m)$ where $n< m$ is
$n(S)=6$
$P(S)=\begin{Bmatrix}\phi,\{1\}, \ldots\{6\},\{1,2\}, \ldots, \\\{5,6\}, \ldots,\{1,2,3,4,5,6\}\end{Bmatrix}$
$-64$ elements
case $-1$
$f(6)=S$ i.e. 1 option,
$f (5)=$ any 5 element subset $A$ of $S$ i.e. 6 options,
$f (4)=$ any 4 element subset $B$ of $A$ i.e. 5 options,
$f(3)=$ any 3 element subset $C$ of $B$ i.e. 4 options,
$f (2)=$ any 2 element subset $D$ of $C$ i.e. 3 options,
$f(1)=$ any 1 element subset $E$ of $D$ or empty subset i.e. 3
options,
Total functions $=1080$
Case $-2$
$f(6)=$ any 5 element subset $A$ of $S$ i.e. 6 options,
$f(5)=$ any 4 element subset $B$ of $A$ i.e. 5 options,
$f^{\prime}(4)=$ any 3 element subset $C$ of B i.e. 4 options,
$f (3)=$ any 2 element subset $D$ of $C$ i.e. 3 options,
$f^{\prime}(2)=$ any 1 element subset $E$ of $D$ i.e. 2 options,
$f(1)=$ empty subset i.e. 1 option
Total functions $=720$
Case $-3$
$f(6)=S$
$f(5)=$ any 4 element subset $A$ of ' $S$ i.e. 15 options,
$f(4)=$ any 3 element subset $B$ of $A$ i.e. 4 options,
$f(3)=$ any 2 element subset $C$ of $B$ i.e. 3 options,
$f(2)=$ any 1 element subset $D$ of $C$ i.e. 2 options,
$f(1)=$ empty subset i.e. 1 option
Total functions $=360$
Case $-4$
$f(6)=S$
$f(5)=$ any 5 element subset $A$ of $S$ i.e. 6 options,
$f(4)=$ any 3 element subset $B$ of $A$ i.e. 10 options,
$f(3)=$ any 2 element subset $C$ of $B$ i.e. 3 options,
$f(2)=$ any 1 element subset $D$ of $C$ i.e. 2 options,
$f(1)=$ empty subset i.e. 1 option
Total functions $=360$
Case $-5$
$f(6)=S$
$f(5)=$ any 5 element subset $A$ of $S$ i.e. 6 options,
$f(4)=$ any 4 element subset $B$ of $A$ i.e. 5 options,
$f(3)=$ any 2 element subset $C$ of $B$ i.e. 6 options,
$f(2)=$ any 1 element subset $D$ of $C$ i.e. 2 options,
$f(1)=$ empty subset i.e. 1 option
Total functions $=360$
Case $-6$
$f(6)=S$
$f(5)=$ any 5 element suhset $A$ of $S$ i e 6 options
$f(4)=$ any 4 element subset $B$ of $A$ i.e. 5 options,
$f(3)=$ any 3 element subset $C$ of $B$ i.e. 4 options,
$f(2)=$ any 1 element subset $D$ of $C$ i.e. 3 options,
$f(1)=$ empty subset i.e. 1 option
Total functions $=360$
$\therefore$ Number of surch funstions $=3240$