Let $\Phi\colon{}^{<\omega}\omega\to\omega$ be a bijection and let $A=\{\{\Phi(b\restriction n) : n \in\omega\}\mid b\colon\omega\to 2\}$
Prove that A is not a mad family.
I'm not sure how to prove this, I think it has to do with showing that it is not maximal, although I do not know how this would work.
Any help would be much appreciated.
Note that $A$ is given by only focusing on $b\colon\omega\to 2$. Therefore any branch in the tree that has any value other than $2$ will necessarily be almost disjoint from all the members of $A$.
So, for example, $\{\Phi(c\restriction n)\mid n\in\omega\}$, where $c(n)=3$ for all $n$, is almost disjoint from all the members of $A$.
You can still prove this is not maximal, even if $A$ was defined using $b\colon\omega\to\omega$ instead of $\omega\to2$. But here we need a slightly more refined approach.
Think about what other subsets of ${}^{<\omega}\omega$ meet every branch on at most a bounded part.
Specifically, note that a branch is a chain, and if we take an antichain $S$, then for a branch $b$, there is at most a single $n$ such that $b\restriction n$ is in the antichain.