What operation is done first in the following exercise...

Question

Here I have such an exercise: I have to simplify the form of the following expression:$$(p\lor \lnot q)\land(\lnot p \lor q )\lor (p \lor \lnot r)\lor \lnot q$$. I know how to simplify it, but what operation is usually done first in discrete math, $\lor$ or $\land$ ? Or , can I move $p \lor \lnot r$ to stay after $p\lor \lnot q$?Thank you very much.

Answer 1

$\lor$ and $\land$ have equal precedence, so we typically evaluate from left to right.

Here, because of associativity of $\lor$, all disjuncts follwing the $\land$ connective can be grouped together, as follows:

$$(p \lor \lnot q) \land (\lnot p \lor q \lor p \lor \lnot r \lor \lnot q)$$

Having done this, we see that within the grouped disjunctions to the right of $\land$, we have $p \lor \lnot p = T$ present (not to mention the presence of $q\lor \lnot q = T)$, making the entire chain of five disjuncts true, leaving us with with the equivalent statement $$(p \lor \lnot q)$$

Answer 2

Typically, $\vee$ and $\wedge$ are of equal precedence, and so parentheses are used to specify which must come first. For example, $(a\vee b)\wedge c$ is not in general the same as $a\vee(b\wedge c).$ Consequently, $a\vee b\wedge c$ is ill-defined, under the standard rules of propositional calculus.

I suspect, therefore, that your text has introduced another convention for the rules of precedence regarding $\vee$ and $\wedge.$ For example, it may specify a left-to-right order, or it may specify that $\vee$ should precede $\wedge$ or vice-versa.