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?
Best Answer
I think your basic problem here is that you expect negation to produce a "complete opposite", whatever that would mean.
The negation of
ought to be nothing more or less than
If Miguel lacks a cellphone but has a computer it is still not true that he has both. So the negation of "he has both" ought to be true as soon as there is one of them he lacks.
In other words, if you take
as the "negation", then you may be satisfying your intuitive sense of "oppositeness", but you have created a situation where it may be that both the sentence and its "negation" are false -- which is absurd.