How to express the binary operators in terms of (only) $\implies$ and $\lnot$? $$
\begin{array}{c|lcr}
A & \text{B} &  \oplus &  \lnot \operatorname{Id}_2  & \lnot \operatorname{Id}_1 \\
\hline
V & V & F & F & F\\
V & F & V & V & F\\
F & V & V & F & V\\
F & F & F & V & V
\end{array}
$$
I don't know how to proceed with this exercise, how can I express the exclusive or?
I do not know how to relate it to the other equivalences, for instance with the normal or it's just $A\lor B\equiv \lnot A\implies B$, how will it be for the exclusive or?
 A: You can write either $(A\lor B)\land\lnot(A\land B)$ or $(\lnot A\Rightarrow B)\land(A\Rightarrow \lnot B)$. The first comes from the usual interpretation of exclusive or as meaning "$A$ or $B$ but not both $A$ and $B$", and the second one comes from noting that exclusive or is equivalent to $\lnot A\Leftrightarrow B$. Now we just use $P\lor Q\equiv\lnot P\Rightarrow Q$ and $P\land Q\equiv\lnot(P\Rightarrow\lnot Q)$ on either to obtain the expression
$$\lnot[(\lnot A\Rightarrow B)\Rightarrow\lnot(A\Rightarrow\lnot B)].$$
A: I have at first be reassured that a solution exists by the interesting information found here. 
Here is a way to express $a$ xor $b$ (xor = exclusive or) using $\lnot$ and $\implies$: 
$$\tag{1}(b \implies a) \implies \lnot(a \implies b)$$
Honestly, I found it by (computer assisted) trial and error by privilegizing expressions with a certain symmetry.
Is there is a shorter one in terms of number of connectors ? Maybe. This one has the advantage of being very "symmetrical".

Edit 1: Proof of formula (1)
Using the fact that $P\rightarrow Q ≡ (\lnot P\lor Q)$, (1) is equivalent to
$$\lnot(\lnot b \lor a) \lor \lnot(\lnot a \lor b),$$
itself equivalent to:
$$(b \land \lnot a) \lor(a  \land \lnot b)$$
which is (one of) the definition(s) of "xor" operator.

Edit 2 : I had a look to your previous question ($A\iff B$ in terms of $\lnot$ and $\implies$) : you could have used the fact that "xor" is the same as the negation of $\Leftrightarrow$ in order to obtain at once a formula for "xor" operator.
