In this post, I’ll show some examples of how to negate propositions.
As I always recommend, let’s start with the definitions.
Table of Contents
- Definitions
- a) Mei has an MP3 player
- b) There is no pollution in New Jersey
- c) 2+1=3
- d) The summer in Maine is hot and sunny
- Summary
Definitions
Let p be a proposition. The negation of p, denoted by ¬p, is the statement “It is not the case that p.” The proposition ¬p is read “not p.” The truth value of the negation of p, ¬p, is the opposite of the truth value of p.
All the definitions I use are from the book Discrete Mathematics and its Applications by Rosen.
According to the previous definition, it is enough to state, “it is not the case that p”, substituting p for its value (the declarative sentence).
However, you will find out that sometimes, when you express the negation of the proposition in English, if you just write as in the definition, it sounds weird.
So, it is useful to use our knowledge of English, to write the sentence in the way that we usually write (in a simplified way). I’ll explain through the examples.
That’s all we need to solve the following exercises.
What is the negation of each of these propositions?
a) Mei has an MP3 player
It is not the case that Mei has an MP3 player.
The way we actually write it in English will be “Mei does not have an MP3 player”.
b) There is no pollution in New Jersey
It is not the case that there is no pollution in New Jersey.
The way we write it in English will be “There is pollution in New Jersey”.
c) 2+1=3
It is not the case that 2+1=3.
We can write this as “2+1 ≠ 3”.
d) The summer in Maine is hot and sunny
It is not the case that the summer in Maine is hot and sunny.
In this case, writing it the way we usually write in English is a bit tricky. Because this one involves compound propositions (see the ‘and’).
You can read about how to write this sentence in a simplified way in the post negating propositions.
Summary
Always start with the definitions you need.
To write a negation of a proposition in English, you can always use the knowledge you already have to write it in a simplified way.
When you are negating a compound proposition, you should follow other rules. See examples on this link.
Related posts:
- Negating propositions
- Which of these sentences are propositions?
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- Propositional logic Exercise 11. Write these propositions using p and q and logical connectives (including negations)
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