NOTty logic
Have a go at being mathematically negative, by negating these
statements.
Age
16 to 18
Challenge level
Problem
The negation NOT$(P)$ of a statement is true if and only if the statement $P$ is false. A well-constructed negation uses positive language, avoiding the use of the word NOT.
Here are two statements, each with four suggested negations. Which of these are the correct negations and why?
1. A good pet is friendly and furry
A. A good pet is unfriendly and unfurry
B. A bad pet is friendly and furry
C. A good pet is unfriendly or unfurry
D. A bad pet is unfriendly or unfurry
2. That man is lying or I'll eat my hat
A. That man is telling the truth and I won't eat my hat
B. Either that man is telling the truth or I won't eat my hat
C. I won't eat my hat or that man is telling the truth
D. That man is telling the truth and I don't have a hat
Have a go at negating this sentence:
If you don't go to the party and if John goes to the party then I won't go to the party
Finally, try to negate this sentence taken from Lewis Carrol's Jabberwocky from Through the Looking-Glass and What Alice Found There, 1872
Twas brillig, and the slithy toves
Did gyre and gimble in the wabe.
This question is based on an exercise from A Mathematical Bridge (2nd ed), by Stephen Hewson. Published by World Scientific.
Getting Started
Look at the variables in each statement which can either, individually, be true or false.
Then draw a truth table for the statements.
The NOT of the statement will be created by switching 1s to 0s, and vice versa, in the appropriate part of the truth table.
Student Solutions
Patrick from Woodbridge school sent us his thoughts, which were artfully clear, in which he reduced each sentence to a clear logical statement which could then be NOTted:
Using de Morgan's law that NOT(A and B) = NOT(A) OR NOT(B) and assuming that Bad means NOT(GOOD):
1. A good pet is friendly and furry is equivalent to
GOOD = Friendly AND Furry
so
NOT GOOD = NOT(Friendly AND Furry)
Using de Morgan's law
NOT GOOD = NOT(friendly) OR NOT(furry)
so d) A bad pet (NOT good) is unfriendly (NOT friendly) OR unfurry (NOT furry).
2. That man is lying or I'll eat my hat
Let X = Lying OR hat
NOT(X)
= NOT(lying OR hat)
= NOT[NOT(lying) AND NOT(hat)]
= NOT(truth AND NOT(hat))
so A. That man is telling the truth and I won't eat my hat
3. If you don't go to the party and if John goes to the party then I won't go to the party
IF [NOT(you) AND john] THEN NOT(me)
so IF NOT[NOT(you) AND john] THEN NOT[NOT(me)] IF you OR NOT(john) THEN me
So, the answer is:
If you go to the party, or if John does not go to the party, then I will go to the party.
4. Twas brillig, and the slithy toves Did gyre and gimble in the wabe.
Let X = brillig AND gyre AND gimble
Then
NOT(X)
= NOT(brillig AND gyre AND gimble)
= NOT(brillig) OR NOT(gyre) OR NOT(gimble)
so Twasn't brillig, or the slithy toves did not gyre or gimble in the wabe.
Steve used truth tables to work out the negations, making use of NOT(X) is True if and only if X is false
Enumerating the possible combinations for friendly/furry we see that a good pet corresponds to a single row in the truth table. A bad pet is found by negating this.
| Friendly | Furry | Good Pet | Bad Pet |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 1 |
| 1 | 0 | 0 | 1 |
| 1 | 1 | 1 | 0 |
We can see that this corresponds to 'A bad pet is not friendly or not furry' because it has the same truth table values as the Bad Pet column:
| Friendly | Furry | Not Friendly | Not Furry | (Not Friendly) OR (Not Furry) |
| 0 | 0 | 1 | 1 | 1 |
| 0 | 1 | 1 | 0 | 1 |
| 1 | 0 | 0 | 1 | 1 |
| 1 | 1 | 0 | 0 | 0 |
For the second part, it is confusing because going/not going are opposites. To be clear, the statement is
X: If you don't go to the party and if John goes to the party then I won't go to the party.
This is a little ambiguous because it does not say that I will go to the party in all other cases. I will assume that this is the case, which quickly gives the truth table for me going as:
| You go | John goes | You don't go | I go (according to rule X) |
| 0 | 0 | 1 | 1 |
| 0 | 1 | 1 | 0 |
| 1 | 0 | 0 | 1 |
| 1 | 1 | 0 | 1 |
| You go | John Goes | I go (according to rule NOT(X)) |
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 0 |
| 1 | 1 | 0 |
There is only 1 case in which I go now. This is logically the same as If you go to the party, or if John does not go to the party, then I will go to the party.
It is also the same as I won't go if John doesn't go or if you go.
Teachers' Resources
Why do this problem?
The overall goal of this activity is for students to leave with an enhanced understanding of clear mathematical communication which they can apply in all other areas of mathematics. It encourages this clarity of mathematical communication by means of trying to negate statements. This provides a good way for students
to become aware of the ambiguities present in everyday communication and naturally reinforces the need for clarity in more advanced mathematics. It is also conversational, quick and good fun.
Possible approach
This problem works well if approached through group conversation: there is nothing like asking for a verbal explanation to make students realise that they do not understand the concepts or are speaking unclearly. Start with the first statement and simply ask: Which do students think is the correct negation? Various points will arise: what, EXACTLY, does negation mean? How might we tell if a
statement has been properly negated? How can we CLEARLY argue this? Does it matter if the statement is actually true or false in reality? It is important to follow the resulting discussion so that students realise the need for clarity and then attempt to resolve this need for themselves.
The goals of this first part would be:
1) Understand clearly the point that the negation of a statement P is true if and only if the statement P is false.
2) Understand that we are not asking directly as to the truth or falsehood of P or any of its constituent parts.
3) Understand that clarity of communication is very important, and that a truth table is perhaps the clearest way forward.
Once these points have been raised, try to work on the second statements and then, finally, the last compound statement. Throughout, encourage precise, clear mathematical communication: with each explanation ask the rest of the group: Was that explanation clear?
As a collective follow up activity, students could try to invent their own sentences with four suggested negations. This is a really powerful part of the activity, as constructing the questions will encourage full engagement with the underlying concepts.
Key questions
At each stage, are you absolutely sure that your arguments are clear and unambiguous?
Did you think that that was a clear explanation?
Possible extension
The activity involving constructing their own sentences for negation can be taken to complicated levels. Students with a clear grasp of logic might try to flip through a text book, find a theorem and construct the negation of that.
Possible support
Students who initially struggle with constructing the negations are likely to be able to spot flaws in the explanations of others. Initially, such students could be given the role of 'judge' to determine whether a particular argument in favour of a negation is valid.