I think I'm regularly screwing up some very simple logic and inclusion concepts in set theory revolving around
or operators and want to figure out on which points my views are wrong.
$P = \emptyset \land Q = \emptyset$. I've seen a proof showing that this means $P\cup Q = \emptyset$. While my English interpretation is that if they are independently equal to $\emptyset$, it is clear that their union must also $ = \emptyset$. Similarly to how $0 + 0 = 0$. However, mathematically I've been taught to associate $\land$ with $\cap$. Does this mean the statement $P = \emptyset \land Q = \emptyset$ is different from $P \land Q = \emptyset \implies P\cap Q = \emptyset?$
$P = \emptyset \lor Q = \emptyset$. My English interpretation of this is that at least one of $P, Q$ is equal to the emptyset, but it is possible that both are (it still satisfies logical or). We can't bet on both being equal to $\emptyset$, because we're only sure 1 is. The intersection between $(ANYTHING)\cap \emptyset = \emptyset$ so this is the most general guarantee we can make. However I'm tempted to swap $\lor$ out with $\cup$ because I'm taught they are essentially equivalent. Since in this situation it seems they are not I'm lead to believe $P = \emptyset \lor Q = \emptyset$ is different from $P \lor Q = \emptyset \implies P\cup Q = \emptyset$.
I think my confusion with a lot of this comes from the fact that $and$ as well as $or$ don't necessarily act as operators in English text. For example the statement:
- 1.) $A$ and $C$ are disjoint and 2.) $B$ and $C$ are disjoint - in English this directly translates to #1 above, however when writing it out I come up with this $\forall x, x \notin (A\cap C)\cap(B\cap C)$. Am I taking the 2nd "and" too literally, and treating it as an operator? It seems that sometimes "and" means $\cap$ but sometimes not and I'm a bit confused. I think I can make the justification that with this case, "and" does not imply any operation on the two disjoint sets because the statement never says anything about them being disjoint or not. It just talks about each set $P = A\cap C, Q = B\cap C$ as being independently disjoint.
Below on paper exist the breaking down of several logical (or illogical lol) inclusion statements. I have gotten two answers for most of them and can't figure out where my logic is wrong. Any guidance/help in clearing this stuff up ill be very very appreciated! I think I'm greatly over thinking this topic because it seems fairly straight forward and trivial.