so I'm rather new to Discrete Math and I've been trying to tackle propositional logic. I have the following question:
How many situations are there for any given proposition?
This question really confuses me. First of all, by situations I guess they mean outputs, and I guess the conclusion can only ever be one at a time? Or are situations rather a combination of truth values?
Any input is appreciated.
The question is not specific enough, but I have an interpretation which might be what they meant. You have $m$ atomic statements you can use in the condition $A$ and $n$ atomic statements you can use in the conclusion $B$ of a statement, then at most how many logically independent statements (of the form $A\rightarrow B$) can you phrase?
Question is, what can you do with atoms. Again, not all that clear, but one interpretation is that you can disjunct them to obtain formulas. As disjunction is commutative and associative, there are $2^k$ disjuncts you can form out of $k$ atoms. So there are at most $2^m$ possibilities for $A$ and $2^n$ possibilities for $B$. This yields $2^{m+n}$ logically independent statements, at most.
Why at most? Because if there are dependencies between atoms you can use in $A$ and in $B$ (e.g., if some of them coincide), then you get less.
So there are many unclear points around the question, you should ask for clarification. For example, if you can apply any Boolean operation on the atoms, then there are at most $2^{2^m}$ possible $A$ and $2^{2^n}$ possible $B$, which yields potentially $2^{2^m+2^n}$ statements altogether.