Sheffer’s functor and Peirce’s functor

Question

Define the connectives: negation, disjunction, conjunction and implication in terms of

a) Sheffer’s functor |, where p | q ⇔ ∼ p ∨ ∼ q

b) Peirce’s functor ↓, where p ↓ q ⇔ ∼ p ∧ ∼ q.

What the idea of the task?

Answer 1

You need to find an expression that only uses the Sheffer functor that is equivalent to $\neg p$, and likewise for $p \land q$, $p \lor q$, etc.

For example, the expression $p|p$ is equivalent to $\neg p \lor \neg p$, which is equivalent to $\neg p$. And so there you go: you can rewrite $\neg p$ as $p|p$

How about $p \land q$? Well, by DeMorgan we know that $p \land q$ is equivalent to $\neg (\neg p \lor \neg q)$ and thus to $\neg (p|q)$ ... and since we just saw that you can write $\neg p$ as $p|p$, that means that $\neg (p|q)$ is equivalent to $(p|q)|(p|q)$. In sum: $p\land q$ can be expressed as $(p|q)|(p|q)$

Ok, so now you try the others!

Answer 2

The idea of the task is for you to write

$$p \implies q $$

In terms of those two functors, as well as

$$\tilde{} p $$

And

$$p \vee q, p \wedge q $$

For example, note that

$$\tilde{}(p \wedge q) = \iff \tilde{}p \vee \tilde{}q $$

And hence

$$p\wedge q = \tilde{}(p|q) $$