A problem with a boolean function.

Question

This is one of my assignment problem and I almost finished others. This question confused me for quite a long time and now I get nothing about that. I did understand what is an increasing function but that made no sense up to now.

Answer 1

I will untangle the problem statement a bit and give a few hints:

A function is increasing if whenever $y$ contains more 1's than $x$, if the function is mapping $y$ to $0$, it also maps $x$ to $0$, so that $f(x) \le f(y)$. Now, there are $2^n$ sequences in $[0, 1]^n$, and for each sequence, if a given function maps it to 0, then all the sequences with a smaller quantity of 1's than it will also get mapped to 0.

There are exactly $\binom{n}{k}$ sequences $x$ so that $|| x|| = k$ for each $k = 0, 1, ..., n$

From that you should be able to count how many functions with the required conditions exist.