Sheffer’s functor and Peirce’s functor 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?

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

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

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