So I am dealing with proving completeness of a set of connectives, and I always struggle with finding equivalent forms of connectives. It seems to take a lot of bruteforcing (which I despise) for me to generate equivalent forms.
For example, let $\underline{0}$ be a unary operator turning anything $0$ (False). Define $\land$, $\lor$ and $\neg$ in terms of only connectives in $\{\rightarrow, \underline{0}\}$.
I attempted to write down basic combinations and try to find a pattern, but this only worked for $\neg A$ which can be defined $A \rightarrow \underline{0}A$. I am not necessarily interested in the answers to $\land$ and $\lor$, but rather, I am interested in strategy.
Best Answer
I am afraid that with this kind of question there is always a little guesswork involved. But, you successfully found the negation, and now you can rely on known equivalences such as $ P \to Q \Leftrightarrow \neg P \lor Q$ (Implication) and DeMorgan to find formulas for $\lor$ and $\land$. They may not be the most efficient ones, but at least those you would get without any further trial and error.
Ok, but you still had to find that formula for the negation, and isn’t that annoying? Well, I don’t think you should get too annoyed with this … I think many if not most mathematical results are found by first ‘playing around a little bit in whatever sandbox you find yourself in … and once you do, certain patterns and ‘moves’ will quickly reveal themselves.