Propositional Logic - Associative Property question

Question

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

Is this move "legal"? Or can you only apply the associative property on like operators?

Answer 1

Associativity applies only when the connectives involved are exclusively $\land$ or exclusively $\lor$:

$$p \land q \land r \equiv (p \land q)\land r \equiv p \land (q\land r)$$

$$p \lor q \lor r \equiv (p \lor q)\lor r \equiv p \lor (q\lor r)$$

Because of associativity of $\lor$ and $\land$, parentheses are not necessary to define expressions like those above.

Your statement, however:

$$p \land \lnot q \lor q \land \lnot r \lor \lnot p \lor r \tag{given}$$

has mixed connectives, and so associativity does not apply across all possible groupings.

Please note: as stated, your (given) expression is not well-defined without parentheses. That is, without parentheses, it is ambiguous; it can be read any number of ways, most of which are not equivalent. Does it mean connect from left to right?:

$$(((((p\land \lnot q) \lor q) \land) \lnot r)\lor\lnot p) \lor r\;?\tag{1}$$

Or does it mean this?

$(p \land \lnot q) \lor (q \land \lnot r) \lor (\lnot p \lor r)\;?\tag{2}$

or any number of other possible ways of grouping with parentheses?

---

In general, when you have an expression like $(2)$ above, you need to apply the Distributive Laws to distribute over another connective:

For example $$p \land (q \lor r) \equiv (p \land q) \lor (p \land r)$$ $$p \lor (q\land r) \equiv (p \lor q) \land (p\lor r)$$

Answer 2

No, mixed expressions like this are not associative; instead, they obey distributive laws: $$ (a\wedge b)\vee c \equiv (a\vee c) \wedge (b \vee c) $$ and $$ a \wedge (b\vee c) \equiv(a\wedge b) \vee (a \wedge c). $$