A fun problem - find the formula of number of functions from a power set to another set.

Hi people! This is yet another problem from the entrance test to CMI (Chennai Mathematical Institute) (http://www.cmi.ac.in/).

Consider sets $\displaystyle A = \{1,2,...,k\}$ and $\displaystyle B = \{1,2,...,n\}$ . Denote $P_k$ as the power set of $A$ . How many functions $f$ can be defined from $P_k$ to $B$ such that $f( M \cup N ) = \text{max} ( f(M), f(N) )$ ?

Example: For $k = 2$ , this function is valid:

$f( \phi ) = 2$
$f(\{1\}) = 3$
$f(\{2\}) = 5$
$f(\{1\} \cup \{2\} ) = f( \{1, 2 \} ) = \text{max} ( f(\{1\}), f(\{2\}) ) = 5$

While the following function is invalid:

$f( \phi ) = 2$
$f(\{1\}) = 3$
$f(\{2\}) = 5$
$f(\{1\} \cup \{2\} ) = f( \{1, 2 \} ) = 3$

Your answer must be a formula involving $n, k$ only. For $n = 4, k = 3$ , the number of such functions is $100$ .

I had fun solving it!

#Functions #Sets #JustForFun #SetTheory #Cmi

Easy Math Editor

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Comments

Is it $\displaystyle \sum_{i=1}^n i^k$ ?

Siddhartha Srivastava - 7 years ago

That it is! Great!

Parth Thakkar - 7 years ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$