Example of use De Morgan Law and the plain English behind it.

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?

Solution 1:

I think your basic problem here is that you expect negation to produce a "complete opposite", whatever that would mean.

The negation of

Miguel has a cellphone and a computer.

ought to be nothing more or less than

It is not true that Miguel has a cellphone and a computer.

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

Miguel has neither a cellphone nor a computer.

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.

Solution 2:

I think a different, more concrete example could make it easier to understand in plain English.

Let's say you have to take an exam which consists of 2 parts, a written test and an oral test. You get a mark on both, and they are independent, which means that you can pass both parts, or fail both, or pass one part but fail the other (doesn't matter which).

The rules are that to pass the full exam you must pass the written test AND the oral one.

Now, you come to me and say:

"I've passed the exam."

What does it imply? What can I infer? Clearly, that you passed the oral test AND you passed the written test. So far so good.

If, instead, you come to me and say the opposite:

"I've failed the exam."

What can I infer? Careful here: I can only infer that you failed the written part, or you failed the oral part, or both. Mathematically, if we use "or", there's no need to add the "or both" at the end, so we can shorten it to this: you failed the written part OR you failed the oral part.

This is what De Morgan's law tells us: the negation of "passed oral part AND passed written part" is "failed oral part OR failed written part". Failing either part is enough to fail the entire exam, there's no need to fail both parts to be rejected.

If I decided that the negation is the complete opposite, as you say, I could conclude something wrong. For example, if you passed the written test but failed the oral one, you would come to me and say "I have failed the exam".
In that case, if I concluded that you must have failed both the written part AND the oral part (complete opposite), I'd be wrong.

Solution 3:

Then “Miguel has a cellphone and he has a laptop computer” can be represented by p ∧ q.

Yes, which is equivalent to “Miguel has both a cellphone and a laptop computer”.

The negation of that is "Miguel does not have both a cellphone and a laptop computer", which means that he is missing at least one of the two items: he may not have a cellphone, or he may not have a laptop, or he may have neither. That is precisely what the quoted solution says.

we can express the negation of our original statement as “Miguel does not have a cellphone or he does not have a laptop computer.”

Note that the logic "or" is inclusive, so the above also covers the case where Miguel has neither a cellphone nor a laptop.

Solution 4:

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?

That is not the complete opposite since it accounts for fewer cases where the original sentence is not true.

Draw a Venn Diagram of two overlapping circles. The complete opposite of their intersection is everything other than the intersection, not merely everything outside both circles.