The formal difference between $\cup$ and $\vee$ I have a problem with understanding the difference between the meaning of the set-union ($\cup$) operator and the Logical disjunction ($\vee) operator.
Namely what I am not sure about are the differences between the calculation rules for those different operators and whether or not they can be interchanged in some situations???
If one is given a rule, like:
M∩(N∪L) = (M∩N)∪(M∩L)   Is it equivalent to, this?   M∧(NvL) = (M∧N)v(M∧L)
I am asking this because, for instance on Wikipedia the De Morgan's laws, are written as if these symbols could be interchanged:

*

*see: https://en.wikipedia.org/wiki/De_Morgan's_laws
All help is appreciated.
Thank You
 A: The difference is that $\cap$ and $\cup$ and $\overline{\phantom{M}}$ operate on sets, whereas $\land$, $\lor$ and $\neg$ operate on formulas.
$\{a, b\} \cap (\{b\} \cup \{b, c\}) = (\{a, b\} \cap \{b\}) \cup (\{a, b\} \cap \{b, c\}$ makes sense.
$(p \to q) \land (\neg q \lor r) = ((p \to q) \land \neg q) \lor ((p \to q) \land r)$ makes sense.
$\{a, b\} \land (\{b\} \lor \{b, c\}) = (\{a, b\} \land \{b\}) \lor (\{a, b\} \land \{b, c\}$ does not make sense.
$(p \to q) \cap (\neg q \cup r) = ((p \to q) \cap \neg q) \cup ((p \to q) \cap r)$ does not make sense.
That being said, the similarity between the two -- the visual similiarity in the symbols, and the fact that the same patterns of rules apply to them -- is not coincidental: It comes fom the fact that both are defined by the same meta-theoretical operators:
$M \cap N = \{x: x \in M \text{ and } x \in N\}$
$M \cup N = \{x: x \in M \text{ or } x \in N\}$
$\overline{M} = \{x: \text{not } x \in M \}$
$\mathfrak{A} \models P \land Q \Longleftrightarrow \mathfrak{A} \models P \text{ and } \mathfrak{A} \models Q$
$\mathfrak{A} \models P \lor Q \Longleftrightarrow \mathfrak{A} \models P \text{ or } \mathfrak{A} \models Q$
$\mathfrak{A} \models \neg P \Longleftrightarrow \text{not } \mathfrak{A} \models P$
Both definitions mimic in the same way the behavior of natural language "and", "or" and "not" from which they are derived, but they operate on different kinds of mathematical objects -- one on sets, the other on formulas -- and are thus not interchangeable.
