Negation of The Statement "The sides of the triangle $\triangle$ have different lengths"

Question

In exercise 2.13 of page 43 of the book Mathematical Proofs: A Transition to Advanced Mathematics the reader is asked to state the logical negation of some statements. Of these, I find the authors' answer to one of them baffling.

The statement to negate is:

"Two sides of the triangle have the same length."

The authors' negation of the statement is:

"The sides of the triangle have different lengths".

Am I mistaken in assuming that when negating a statement, one is supposed to state what previously presumed false as true and vice versa? If one assumes the proposition "Two sides of the triangle have the same length." to be true, is it erroneous to conclude that the negation would be "The sides of the triangle have different lengths" or (exclusively) 'Three sides of the triangle have the same length'? I thank your aid in advance.

Answer 1

Your difficulty arises from your interpreting the sentence "two sides are equal" as "two sides are equal (and the third side is a different length)". However that is not what "two sides are equal" means. "two sides are equal" means "there are two sides that are equal... we don't know which two sides are equal and we don't know anything about whether the third is or is not also equal to those two sides".

Hopefully if you view it that way you can say why the answer was what it was.

.... read on if you wish......

If all three sides are equal then any two sides will be equal so all three sides is compatible with (and is a subspace of) two sides being equal.

The negation of "two sides are equal" is "there are no two sides that are equal" and that is equivalent to "all sides are different".

However if we take the statement "EXACTLY two sides are equal" that would mean that two sides are equal and the third is a different length. The negation of that would be: That it is not the case that exactly two sides are equal so either there are fewer than two sides that are equal (no two sides are equal) or there are more than two sides that are equal (there are three sides).

So the negation of "EXACTLY two sides are equal" would be "Either all sides are different or all sides are the same".

.....

But "two sides are equal" does NOT mean "exactly two sides are equal". "two sides are equal" means "there exists at least one pair of equal sides". And the negation IS "all sides are different".

Answer 2

Let $x,y$ and $z$ be the sides of a triangle. We have: $x\neq y, ~ x\neq z,~ y\neq z$

Let $len(s)$ be the length of side $s$.

We have two sides being equal in length:

$~~~~~~len(x)=len(y)~ \lor ~len(x)=len(z) ~ \lor ~ len(y)=len(z)$

(EDIT: All three sides being equal in length is not ruled out.)

Applying De Morgan's Law and the elimination of double negations ($\neg\neg$), we can obtain its negation:

$~~~~~~len(x)\neq len(y) ~\land~ len(x)\neq len(z) ~\land ~ len(y)\neq len(z)$

In words, we have NO two sides being equal in length.

Answer 3

I hope it's possible to be logically consistent and easy to understand, I'll try to be both.

I think both versions of the sentence are equivalent, and I'll try to prove it, or at least to convince you.

Hypothesis: The sides are of different length. $ \Leftrightarrow$ No two sides are of equal length.

Proof: let $a,b,c$ be the lengths of the sides.

Now let all three numbers be different: $\Leftrightarrow a\neq b \land a\neq c \land c \neq b $ that is obvious right? This obviously implies that no two sides should be equal.

Now we look at the ops idea and say: Let no two of the numbers be equal.

This is not as intuitive I think, but I hope that it will become clear.

We start by saying $ a\neq b $ so far so good but what about $c$? It can't be equal to $a$ because then two numbers would be equal, so we need to add that with a logical and because the inequalies must both be fulfilled, one or the other would not be enough: $\ a\neq b \land a\neq c$. Now it is getting more obvious right? $c$ also can't equal $b$ and again we need that to be true at the same time.

So the result is: let no two of them be equal $\Rightarrow a\neq b \land a\neq c \land c \neq b $.

Concluding, that no two sides being equal implies all sides being different and vice versa.

We could have started with any two of the sides and would have had to add the other inequalies.

Answer 4

"Two sides of the triangle have the same length."

In other words, the triangle is isosceles (possibly equilateral).

So, the statement's negation is

• Each side of the triangle has a distinct length.
• Each side of the triangle has a different length (from the other two).
• The triangle's sides have different lengths from one another.

A formalisation of the given sentence: $$\exists(x,y)\;(Lxy\land x\ne y);$$ its negation: $$\forall (x,y)\;(\lnot Lxy\lor x=y).$$

the negation would be "The sides of the triangle have different lengths"

Correct.

or (exclusively) 'Three sides of the triangle have the same length'?

Incorrect. Every equilateral triangle is automatically isosceles; that is, if a triangle is equilateral, then it does have two sides with a common length, which is not saying that it has exactly two sides with a common length.

After all, if I I have exactly 10 fingers, then

• “I have 2 fingers” is absolutely true (albeit misleading);
• “I have at least 2 fingers” is also true;
• “I have exactly 2 fingers” is false.

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Addendum

Rosie F:

@fleablood "we should... teach the students that statements only imply what they explicitly state." That principle seems reasonable. So how is it consistent with your contention that the statement means "at least two", and that to interpret it as "exactly two" is wrong? The statement explicitly stated "two", and did not state "at least" (or any other modifier). It seems that students need to learn both to make and to understand the difference between the statements.

I'm not fleablood, but the point is that ‘at least two’ is the most conservative interpretation: notice that ‘exactly two’ means ‘at least two and at most two’, which contains an unwarranted assumption that restricts the possibilities.