And here’s the post I wrote today:
First, we note that by definition. Since our complementation reverses order, we find . Similarly, . And thus we conclude that .
On the other hand, by definition. Then we find by invoking the involutive property of our complement. Similarly, , and so . And thus we conclude . Putting this together with the other inequality, we get the first of DeMorgan’s laws.
To get the other, just invoke the first law on the objects and . We find
Similarly, the first of DeMorgan’s laws follows from the second.
Interestingly, DeMorgan’s laws aren’t just a consequence of order-reversal. It turns out that they’re equivalent to order-reversal. Now if then . So . And thus .