Logic Pure Subset Problem

Question

for example if we define :

$$ \$(p,q,r) = (p\to q)\land(\neg p\to r)$$

how we can inference that set $\{\$,\top,\bot\}$ is Full Functional and not any pure subset of this be full functional.

Answer 1

For some general remarks on how to prove that sets of connectives are/are not complete, see How to prove that a set of logical connectives is functionally complete(incomplete)?

In this particular case, it is fairly easy to figure out how to build $\vee$ and $\neg$ out of $\{\$,\top,\bot\}$, once you see that $\$(p,q,r)$ equals $q$ if $p$ is true and $r$ otherwise.

On the other hand, to see that there is no complete proper subset, it is enough to see that every subset with just one connective removed is incomplete.

For example, one can see that every expression built from $\{\$,\top\}$ alone has the property that it evaluates to true when all inputs are true, so in particular $\neg$ cannot be realized. The cases for $\{\$,\bot\}$ and $\{\top,\bot\}$ are similarly simple.