De Morgan's Law is often introduced in an introductory mathematics for computer science course, and I often see it as a way to turn statements from AND to OR by negating terms.
Is there a more intuitive explanation for why this works rather than just remembering truth tables? To me this is like using black magic, what's a better way to explain this so that it makes sense to a less mathematically inclined individual?
Insert real-world predicates and read aloud, for instance:
It can not be both winter and summer (at any point in time).
and
(At any point in time) It is not winter or it is not summer.
Clearly, the two statements are equivalent.
If you like to visualize it, use the venn diagrams. See this, for instance.
I find it more simple just to memorize the basic 2 laws: everytime you "break" a negation line, you replace the AND to OR (or vice versa). Adding two negation lines changes nothing (but gives you more "lines" to break). It just works.
The statement $\left(\bigcup_i A_i\right)^c \subseteq \bigcap_i A_i^c$ is equivalent to $$x \in \left(\bigcup_i A_i\right)^c \Longrightarrow x \in \bigcap_i A_i^c$$ and can be read as follows:
If $x$ is not in some $A_i$, then $x$ is not in any $A_i$.
I think this latter statement is obvious. You can similarily read the converse inclusion.
$\mathbf{(A \cdot B)'} = A' + B'$
They can't make you pizza and chips:
They're either missing the pizza, missing the chips or missing both
$\mathbf{(A+B)'} = A'\cdot B'$
He doesn't have an Xbox or a Playstation
He doesn't have an Xbox and he doesn't have a Playstation
