I am currently reading "Discrete Mathematics and Its Applications, 7th ed", p.29.
Example:
Use De Morgan’s laws to express the negations of “Miguel has a cellphone and he has a laptop computer”.
Solution:
Let p
be “Miguel has a cellphone” and q
be “Miguel has a laptop computer.” Then “Miguel has a cellphone and he has a laptop computer” can be represented by p ∧ q
. By the first of De Morgan’s laws, ¬(p ∧ q)
is equivalent to¬p ∨¬q
. Consequently, we can express the negation of our original statement as “Miguel does not have a cellphone or he does not have a laptop computer.”
Here and in De Morgan law I think I understand the math part. I am constructing truth tables of propositions and I see why propositions are equivalent in De Morgan law.
But I do not understand plain English part of the example. As I understand complete opposite means as opposite as possible and negation is the complete opposite. Why negation (complete opposite) of "Miguel has a cellphone and he has a laptop computer" is "Miguel does not have a cellphone or he does not have a laptop computer". Why complete opposite is not "Miguel does not have a cellphone and he does not have a laptop computer"? I mean if he does not have both it is more opposite than if he does not have one of them, right. Why is it so?