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In propositional logic, one could not correctly say that : $(A \& B) = (B \& A)$.

The reason is syntactic: the first conjunct of $(A \& B)$ is $A$, while the first conjunct of $(B \& A)$ is $B$. So the two formulas are not identical, they are not the same formula.

The only thing one can say is that the two formulas are equivalent.

My question is: does this syntactic argument also hold for fractions?

Remark. I write this question after having watched a video by Herbert Gross where he expresses his reluctancy to call two fractions like $1/5$ and $2/10$" equal". According to Gross, they would be better called "equivalent" inasmuch as they " name the same number"

Remark. I do not ask whether the equivalence relation between fractions is the same as the logical equivalence relation " formula X is true in exactly the same interpretations as Y ". My question is not :

does " 1/5 = 2/10 mean 1/5 <=> 2/10" ?

I simply ask whether the equal sign between fractions should be read as some sort of arithmetical equivalence ( not a logical one of course).

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    $\begingroup$ The construction of $\Bbb Q$ as the fraction field of $\Bbb Z$ does exactly this. It consists of equivalence classes. $\endgroup$ Apr 27, 2019 at 10:47
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    $\begingroup$ they are equal in value but equivalent in notation. $\frac{1}{5}$ does not look like $\frac{2}{10}$ at all. and as Burde said, it's construction. $\endgroup$
    – user29418
    Apr 27, 2019 at 10:48
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    $\begingroup$ To reference Morris Kline's Mathematics for the Nonmathematician, (page 62) he describes that in order to add the fractions $2/3$ and $7/5$, one must "express each fraction in an equivalent form such that the denominators are now alike..." $\endgroup$
    – WaveX
    Apr 27, 2019 at 10:50
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    $\begingroup$ They are two names of the same rational number. We use them in an expession like $\dfrac {2}{10} \times 5 = \dfrac 1 5 \times 5$ according to the axiom fro equality : $x=y \to (f(x)=f(y))$. $\endgroup$ Apr 27, 2019 at 11:03
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    $\begingroup$ There is some confucion here between logical symbols and set-theoretical/arithmetical ones. In logic language, "logical equivalence", i.e. bi-conditional, is a symbols expressing a relation between formulas, while equlity is a relation between terms. Thus $1+1=2$ and not $1+1 \leftrightarrow 2$. $\endgroup$ Apr 27, 2019 at 11:12

3 Answers 3

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Whether $2/10$ and $1/5$ are equivalent or equal depends on how you define the meaning of the formal expression "$a/b$".

If $a/b$ is just a convenient way to write the ordered pair $(a,b)$ of integers when you are discussing the rational numbers, then those two fractions are equivalent - they define the same rational number.

If $a/b$ is just a way to write the rational number that solves the equation $bx=a$ then those two fractions are equal.

In an application they may not even be equivalent. Kids are taugh to model "$1/5$" as "cut a pie in $5$ parts and take $1$ of them". That is not the same physical operation as "cut a pie in $10$ parts and take $2$". That lack of equivalence is even clearer for the commutativity of multiplication: two kids with three cookies each is not the same as three kids with two cookies each even though the number of cookies is the same.

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Construction of the field of rational numbers $\mathbb{Q}$, from the integers $\mathbb{Z}$, may help. In this construction, we deal with equivalent classes $[(a,b)]$ for $b\ne 0$, defined by $$[(a,b)]=\{(x,y)\in\mathbb{Z}\times(\mathbb{Z}-\{0\}): ay=bx\}. $$

Therefore, for example, the fraction $\frac{1}{5}$, is equal to the class $[(1,5)]$ which also contains infinitely many elements like $(2,10),(3,15),\ldots$ and all of this elements are a representation of the same class, namely $\frac{1}{5}$.

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  • $\begingroup$ Yes, what's important to note is that although the couples $(1,5)$ and $(2,10)$ are merely equivalent, the fractions $\frac{1}{5}$ and $\frac{2}{10}$, which are equivalence classes, are literally equal $\endgroup$ Apr 27, 2019 at 11:42
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$\dfrac2{10}$ and $\dfrac15$ denote the same rational number. As rationals, they are equal. As fractions, up to your taste.

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