I recently posted a question with a similar title, but reading the community guidelines, I decided to improve it :)
We can define the $AND$, $OR$ and $NOT$ operations in terms of the $T$ and $F$ operators and the standard combiners as follows:
$K \equiv \lambda xy.x$
$S \equiv \lambda xyz.xz(yz)$
The logical operations will then be
AND (pós-fixed): $AND \equiv F \equiv SK$
Ex: $(TT)(AND) \equiv TTF \equiv T(TF) \equiv T$
OR (in-fixed): $OR \equiv T \equiv K$
Ex: $(T)(OR)(T) \equiv TTT \equiv T(TT) \equiv T$
NOT (pós-fixed): $NOT \equiv FT \equiv (SK)K$
Ex: $T(NOT) \equiv TFT \equiv T(FT) \equiv F $
$p \to q \equiv \neg p \lor q \equiv (pFT)Tq \equiv p FTT q$
So the conditional is an in-fixed operation.
With all that said, let's get to the problem.
Define a lambda term $X$ that corresponds to the logical “or exclusive” operation, ie, $Xpq = T$ if $p$ and $q$ have different logical values and $Xpq = F$ otherwise.
In other words, the problem wants a prefixed description of the XOR operation in terms of the lambda calculus
$p \veebar q \equiv (p \lor q) \land \neg(p \land q) \equiv (p T q)((pqF)FT)F $
Right now I can't follow up to produce something like $Xpq$...any tips? Since $X$ has to be described in terms of $T$ and $F$.