De Morgan's Law - Propositional Calculus
Can someone tell me what's the correct way to solve this proposition with De Morgan's Law?
Proposition: $q \vee ¬[ (p \wedge q) \vee ¬q ]$
[Answers]
Option 1:
$q \vee ¬(p \wedge q) \wedge q$
Option 2:
$q \vee [¬(p \wedge q) \wedge q ]$
As you can see, the difference between both is that one of them don't have square brackets "[ ]". So, what's the correct one?
On
Because of how the original expression you can use replacement of variables and rewrite it as $q \vee \neg A$ where A is the expression in braces, because of this idea option 2 is the correct one
On
If you consider $\land$ as having higher precedence that $\lor$ (just like $\cdot $ has higher precedence than $+$), then $$ q \vee (¬(p \wedge q) \wedge q )\equiv q \vee ¬(p \wedge q) \wedge q $$ in the same way as $$ q + \left((p \cdot q)^2 \cdot q \right)=q + (p \cdot q)^2 \cdot q.$$ If you do not define such precedence, then the extra brackets are needed (as would be differently placed brackets for any alternative interpretation).
The second version is correct because you are just applying DeMorgan to the second part. You can simplify it further, but I don't see that as part of the question. If the first $q$ were $r$ throughout, it would matter how you grouped the three terms at the end.