The ability of a logical statement to represent a two-place truth function. How can i determine which two-place truth functions can be represented using a logical statement built out of a subset of two logical connectors in $ \{\rightarrow, \wedge, \vee ,\equiv \}$ ?
for example $\{\rightarrow, \wedge\}$
 A: For any two place truth function X, we can write it's truth table as follows:
p   q   X(p, q)
0   0   ?1
0   1   ?2
1   0   ?3
1   1   ?4

where, of course, ?1, ?2, ?3, and ?4 belong to {0, 1}.  Notice that all wffs of propositional logic can get built up from the variables and the connectives.  For example (using Polish notation) the wff CKpqNDrArs can get built up via the sequence (p, q, Kpq, r, r, s, Ars, DrArs, NDrArs, CKpqNDrArs).  Thus, we can build up any wff using say two (or 1 or 3 or 4) connectives in this way, and see how their truth tables work and see if the columns of the truth tables end up repeating or if we get new columns.  For instance... if we just have implication "C", we can write
p  q   Cpq  CpCpq Cpp  Cqp  Cqq  CqCpq CCpqp  CCpqq  CCpqCpq 
0  0   1    1     1    1               0      0      
0  1   1    1     1    0               0      1
1  0   0    0     1    1               1      1
1  1   1    1     1    1               1      1

I've left some blank, since they have duplicate values to something else we already have.  I generated this example (by hand) by using (p, q, Cpq) as the initial set and finding all possible substitutions in Cxy from that set.  Then leaving only those columns which are not a duplicate of some other column, we can use each wff above the column to see if we get a new column not in our list.  Eventually, since the number of possible columns is finite, we can eventually see which truth functions can get represented (and we also could prove that using the set of all formulas above the column, as I used the initial set above, that on the next step we'll only get a column that we already have.  Since this procedure can generate all formulas in two variables, it will follow that no other binary truth-function can get generated using just the connectives we've selected). 
