Question 136
Let $n \gt 1$ be an integer and $\Omega := \{1, 2, \dots, 2n-1, 2n\}$ the set of all positive integers that are not larger than $2n$.
A non-empty subset $S$ of $\Omega$ is called *sum-free* if, for all elements $x, y$ belonging to $S$, $x+y$ does not belong to $S$. We allow $x=y$ in this condition.
Prove that $\Omega$ has more than $2^n$ distinct sum-free subsets.
A non-empty subset $S$ of $\Omega$ is called *sum-free* if, for all elements $x, y$ belonging to $S$, $x+y$ does not belong to $S$. We allow $x=y$ in this condition.
Prove that $\Omega$ has more than $2^n$ distinct sum-free subsets.