I don't understand how to work backwards to work out a truth table that has been filled out already (I don't know the logical operators). E.g
I need to find the logical operators and connectives (and, or, not, implies etc.) that is equivalent to the output. Please guide me through the steps or teach me how to work out how to do this. Thanks.
Generally speaking, there is no unique boolean function $f(p, q)$ that produces your output. For your example, the function $f(p, q) = p$ would be a valid solution, as well as $f(p, q) = (p \vee q) \wedge (p \vee \neg q)$, and $f(p, q) = (p \wedge q) \vee (p \wedge \neg q)$. You can find an equisatisfiable function to the one that produced the given output, and there are a number of approaches to do so.
A simple approach would be the enumeration of variable constellations that produce $\top$, and disjunct them. For your example that would mean first finding the constellations that produce $\top$:
and finally producing $f(p, q) = (p \wedge q) \vee (p \wedge \neg q)$
This way you produce a function in disjunctive normal form (DNF). A similar approach works for CNF, but in both cases you end up with a fairly large function.
More complex approaches are Karnaugh Maps and the Quine McCluskey algorithm. They can produce much more concise, even minimal equisatisfiable functions.
As far as I know, most approaches focus on CNF (and sometimes DNF), i.e. use only conjunctions, disjunctions and negations, and don't nest operators any further.
The formula that produces your truth table is: $P$ (which you might also be calling $A$). More generally, you can produce the CNF and DNF representations, which in this case are $(P \lor Q) \land (P \lor \lnot Q)$, $(P \land \lnot Q) \lor (P \land Q)$ respectively. There is an algorithm for producing them that you might have learned in class.
