Converting to complements

Question

Let's say I wanted to convert the ands and positive variables to their complements and ors.

Would this be correct?

$$DE=$$ $$(DE)''=$$ $$(D'+E')'$$

Or another example:

$$D'E=$$ $$(D'E)''$$

Can you break the last one down further?

Answer 1

The first case is right. In your second case, just apply the same steps: $$D'E=(D'E)''=((D')'+E')'=(D+E')'$$ I always think about such problems in natural language: what does it mean for both $E$ to hold and $D$ to fail? Well, simply neither for $E$ to fail or $D$ to hold, of course, which is exactly the interpretation of $(D+E')'$.