help with Propositional Logic?? i have to do an exercise for the university , i did that but i'm so sure that is right so: • “The tumor is benign” A • “The tumor is metastatic” B • “The tumor has a good prognosis” C Express the following statements in Propositional Logic: 1\. The tumor has a good prognosis only if it is either benign or not metastatic. 2\. A tumor cannot be benign and metastatic at the same time. 3\. If the tumor is not metastatic, it is either benign or has a good prognosis. To my opinion the result should be: 1.A XOR ¬B ---> C 2.¬A AND B 3.¬B--->A XOR C i read in Internet that i can convert either..or.. with the logical operation XOR but i'm not sure that my result is right because as second question is asked which of these formulas are equivalent, but i tried to do the true table and no one has the same result. Can everyone help me please?

For 1. "P only if Q" translates to $P \rightarrow Q$. The 'only if' works differently from a plain 'if': the 'if' expresses a _sufficient_ condition, while the 'only if' expresses a _necessary_ condition. So, "P only if Q" expresses that Q is necessary for P to be the case or, equivalently, that if Q is not the case, then P is not the case, i.e $\
eg Q \rightarrow \
eg P$ ... which by contraposition is equivalent to $P \rightarrow Q$

For 2. "Not both P and Q" says that it is not true that both P and Q are true, so that translates to $\
eg (P \land Q)$

For 1. And 3. Contrary to whatever source on the internet you were using, you cannot assume that "either ... or..." always translates to an XOR. For example, when I say "If I grow old, I want to be either rich or happy" I certainly don't mean to say that I don't want to be both rich and happy! Context and common sense will often tell you whether an inclusive or exclusive or is meant.