Exception to the Reflexive Property of Equality

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austint

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Hello everyone,

Today, my Geo-Trig teacher gave us a challenge. He said that we must find an exception to the Reflexive Property (a = a) where both sides of the equal sign are the same but they are not equal. He claims that there are upwards of 50 possible answers (though I'm not sure if this is entirely correct or he was exaggerating). He also said that in his 20 years of giving the challenge out, only 3 students could figure it out with two completely different answers. I am completely baffled by the claim that the reflexive property could have an exception. Do any of you have any ideas? Any help is appreciated.

Thank you!
 
Must "a" be a number? Perhaps you are thinking too narrowly.
 
man = woman

Equal, but not the Same :)

Just trying to get some thinking going.

You need "Same, but not Equal".
 
While thinking about this exercise, I began to wonder whether it's a philosophical matter. (I'm still not sure.)

Is a Real value approached in a limit statement the same as the value itself? They are certainly equal, but is each side the same thing?

\(\displaystyle \displaystyle \lim_{x \to 0} \; [2x + 4] = 4\)

How about ∞ = ∞ ? There are different sizes of infinity.

How about a cloned mammal and it's source? Their DNA is identical (same quantity of A,G,C,T pairs), but they are not the same organisms.
 
mmm4444bot said:
While thinking about this exercise, I began to wonder whether it's a philosophical matter. (I'm still not sure.)

Is a Real value approached in a limit statement the same as the value itself? They are certainly equal, but is each side the same thing?

\(\displaystyle \displaystyle \lim_{x \to 0} \; [2x + 4] = 4\)

How about ∞ = ∞ ? There are different sizes of infinity.

How about a cloned mammal and it's source? Their DNA is identical (same quantity of A,G,C,T pairs), but they are not the same organisms.
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My teacher denied all of these. The square root of 4 = the square root of 4, Arctan(1) = Arctan(1), and other situations that have multiple solutions are not the correct answer. It is not a philosophical matter, he says. Rather, there are two numerical quantities in this circumstance.
 
austint said:
My teacher denied all of these.
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We'll need to keep trying.

How about scalars versus vectors?

Vectors are numbers that have a directional component, in addition to a magnitude (size).

A unit vector has magnitude 1; that's equal to the size of the Natural number 1. Therefore, each number is 1, yet they are not the same (one is a vector and the other is a scalar).


The square root of 4 = the square root of 4 … and other situations that have multiple solutions …
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Be careful. When we write the radical:

\(\displaystyle \sqrt{4}\)

it represents only one number: 2

It is true that (-2)(-2) also equals 4, but the radical above represents what we call the "Principle Square Root", and that's always the non-negative root.

-2 is the opposite of the square root of 4. Hence, if you want to talk about -2 as a root of 4, you need to write it as:

\(\displaystyle -\sqrt{4}\)

When you wrote about "multiple solutions", you were probably thinking of solving an equation:

\(\displaystyle x^2 = 4\)

There are two solutions: the square root of 4 and its opposite. The square root of 4 is never negative. The opposite of the square root of 4 is always negative. Again, this is because "the square root of x" refers to the non-negative root. 8-)

Taking the square root of each side, we get:

\(\displaystyle \sqrt{x^2} = \sqrt{4}\)

\(\displaystyle |x| = 2\)
 
I asked him today. He said that it does not have to do with vectors/scalars.
 
Oops -- I was turned around earlier. I should have paid more attention to tk: same, but not equal. :oops:
 
Bob, that's interesting. I haven't seen an overbar used to denote repeating digit(s), to the left of a decimal point.
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Are you doing arithmetic with infinity?
 
Here's my next go (at least my brain waited until eight o'clock, to awaken me this time).

It has to do with the difference between a fraction and a ratio.

2/3 = 2/3

On the left is a fraction. It represents part of a whole (eg: my basketball team won 2 out of 3 games).

On the right is a ratio. It represents a comparison of two numbers (eg: there are 2 juniors and 3 seniors, on my team).

Both sides look the same, but the interpretations are not equal.
 
I have one, but you won't believe it.

In older version of some parsing computer languages (think 1960s), it was inadvertently possible to reassign just a little too much. For example, some character symbols could be treated as variables - including some single digits.

Thus, one could do this:

5 <== 7 -- Meaning, the variable 5 is assigned the value 7.

5 = 7 ==> True.

5 = (2+3) ==> False

There you go. I expect this variation is NOT in the list of 50. :)
 
Ordinarily we only use the equals sign for an equivalence relation. That's a relation that's reflexive, symmetric, and transitive. So in particular such relations must be reflexive by definition.

There are certainly non-reflexive relations though. Consider the relation \(\displaystyle \sim\) on the set of all people where \(\displaystyle a\sim b\) if and only if \(\displaystyle a\) is a sibling of \(\displaystyle b\). Then \(\displaystyle \sim\) is symmetric and transitive, but irreflexive. Usually though, we would not use the equals sign for such relations.
 
Hi

mmm4444bot said:
Bob, that's interesting. I haven't seen an overbar used to denote repeating digit(s), to the left of a decimal point.
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34.0

Are you doing arithmetic with infinity?
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Yes, I am doing arithmetic with infinite digits to both the right and to the left.

10-adic numbers share the same arithmetic with Decimal numbers (are the same in that respect)
10-adic numbers have a different metric than Decimal numbers (are different in that respect)

The measure of a 10-adic number is not in fact infinite, but is different than the Decimal number that behaves the same Arithmetically.

PS: A VERY common use of 2-adic numbers are integers in computer science. These are called "2's Complement".
 
Your example here is correct, but was not accepted by the professor because it is not over the real numbers. Also, he said that there are an infinite amount of possible answers, so it could not be finite (this is not pertinent to your response but I'm just adding more info on the problem). But, congratulations since your answer was the closest so far. We just need it in the real number area.
 
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