De Morgan’s Laws:
¬(p∨q)≡¬p∧¬q
¬(p∧q)≡¬p∨¬q
Table of Contents
- a) Jan is rich and happy
- b) Carlos will bicycle or run tomorrow
- c) Mei walks or takes the bus to class
- d) Ibrahim is smart and hard working
a) Jan is rich and happy
The way to solve this type of exercise and make sure not to make a mistake is straightforward:
- Use propositional variables
- Apply the De Morgan Law.
Let p=”Jan is reach” and q=”Jan is happy”
The negation of p^q is as follows:
¬(p∧q)≡¬p∨¬q, by the De Morgan Law.
Now, we can write it in English.
The negation of the statement is Jan is not rich, or Jan is not happy.
It is a common mistake to state that the negation of the previous statement is “Jan is not rich and is not happy”. Others will answer “It is not the case that Jan is rich and happy”. Both of the previous statements are considered wrong answers, as they can be subject to different interpretations. In other words, they are ambiguous.
As a rule of thumb, do not use “it is not the case” to negate compound propositions.
b) Carlos will bicycle or run tomorrow
Let p=”Carlos will bicycle tomorrow” and q=”Carlos will run tomorrow”.
¬(p∨q)≡¬p∧¬q, by the De Morgan Law (don’t forget to justify every step).
The negation of the statement is Carlos will not bicycle tomorrow, and Carlos will not run tomorrow.
c) Mei walks or takes the bus to class
Let p=”Mei walks to class” and q=”Mei takes the bus to class”.
¬(p∨q)≡¬p∧¬q, by the De Morgan Law.
The negation of the statement is Mei does not walk to class, and Mei does not take the bus to class.
d) Ibrahim is smart and hard working
Let p=”Ibrahim is smart” and q=”Ibrahim is hard working”.
¬(p∧q)≡¬p∨¬q, by the De Morgan Law.
The negation of the statement is Ibrahim is not smart, or Ibrahim is not hard working.
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