Let X and Y are prepositions.
Tautology means Every row of X is 1, if X is a Tautology represent by T(X) ===> ~T(X) represent X is not a Tautology but T(~X) represents '~X' is a Tautology.
Contradiction means Every row of X is 0, if X is a Contradiction represent by C(X) ===> ~C(X) represent X is not a Contradiction but C(~X) represents '~X' is a Contradiction.
Contingency means Preposition is neither Tautology nor Contradiction. ===> ~T(X) ^ ~C(X)
C) The disjunction of two contingencies can be a tautology.
note that disjunction of two contingencies are not always a tautology it may be a contradiction or contingency.
which means there exist two contingencies which disjunction is tautology ===> X is a Contingency and Y is Contingency but their disjunction is a Tautology
represent two different contingencies ( ~T(X) ^ ~C(X) ) and ( ~T(Y) ^ ~C(Y) ) and their disjunction is Tautology T( X V Y )
∴ ∃X ∃Y ( ( ~T(X) ^ ~C(X) ) ^ ( ~T(Y) ^ ~C(Y) ) ^ T( X V Y ) )
but you may confused with ∃X ∃Y ( ( ( ~T(X) ^ ~C(X) ) ^ ( ~T(Y) ^ ~C(Y) ) ) -----------> T( X V Y ) )
the above formula says that every disjunction of two contingencies can be a tautology
D) The conjunction of two tautologies is a tautology
given statement is conjunction of two tautologies is a tautology always.
represent two different tautologies T(X) and T(Y) and their conjunction is Tautology T( X ^ Y )
∴ ∀X ∀Y ( ( T(X) ^ T(Y) ) -------------> T( X ^ Y ) )