Hello,
For the first two...
SIMPLIFICATION VS EXPANSION
Sometimes you simplify and sometimes you expand. When you expand it sometimes leads to more simplifications later or another way to solve something.
NEGATIVE LOGIC
If you take the expression:
A(B'+C)
and invert all inputs and outputs and all logical connectives, you get an equivalent:
A'+BC'
Now you can look at the map and see that this is true so the first representation is correct also.
Logic systems contain a very limited amount of information so there are different ways of looking at things.
Like the old time game, "Asteroids", where the ship would exit one side and enter the other. It's a finite, closed, wrap around world in logic...if you have a '1' then you also know that you dont have a '0' and vice versa. IF A is 1 then it cant be 0, and if 0 then it cant be 1. This leads to simplifications. We can look at the 'cans' or the 'cants' to get some understanding of the system.