I have a statement,
Either p or q
and I have to write it in terms of logical connectives but I don’t get which logical connector should I be using?
Here is what I did (I think there could have been a better way to do this)
$(p \lor q ) \land (\neg((p \Rightarrow q) \land (q \Rightarrow p)))$
To me, the word “either” is unnecessarily confusing, and should be avoided if possible (of course, since it is part of the problem we have no choice in this case).
If “either $p$ or $q$” means the same thing as “$p$ or $q$”, then the answer is simply $p\vee q$ (by the definition of $\vee$).
However, if “either $p$ or $q$” means “either $p$ or $q$, but not both” then this is equivalent to “$p$ is true and $q$ is false, or $p$ is false and $q$ is true”. Do you see how to write the logical expression for this?
If “Either p or q” means “p or q”, then logical disjunction $p \lor q$ would do it, if “Either p or q” means not both, then exclusive disjunction is needed: $p \oplus q = (p\land \neg q)\lor (\neg p\land q) = (\neg p\lor \neg q)\land (p\lor q)$.
There are several symbols for exclusive or, including $\oplus$ and $\veebar$. However, while in classical logic such connectives are both easily defined in terms of existing connectives and by means of a truth-table, they are not commonly employed in mathematics.
It’s worth noting that there are many minimal functionally complete sets of logical connectives, but the definition of xor in terms of formulae using predefined connectives will be different in each. An obvious one is replacing $\phi \oplus \psi$ with $(\phi \vee \psi) \wedge \neg (\phi \wedge \psi)$.
Assuming I understand the question correctly (you’re looking to describe “exclusive or”), hint: either p or q means p is true and not q, or (inclusive or) …
You could also try draw the truth table.