Does symmetric property of Equivalence hold when the left hand side of the equation is in indeterminate form?

Question

The Symmetric Property of Equivalence is a=b implies b=a. This property does not have any conditions on a or b. But what if through manipulation I get $a=\frac00$ where $a$ is a real number does this imply $\frac00=a$? Wouldn't it be a fallacy to define $\frac00$ as anything because it is by definition indeterminate? I have always treated it as a symbol that cannot be manipulated further. Should $\frac00$ be treated as a ratio of $0$ with $0$ or a symbol?

Answer 1

But what if through manipulation I get $a={0\over0}$ where $a$ is a real number does this imply ${0\over0}=a$?

Expression ${x \over y}$, whatever set $x, y$ belong to, usually (that is, unless explicitly stated otherwise) is compact form for $x * y^{-1}$ where $y^{-1}$ is such an element of set that $y*y^{-1}=1$ (said set has to be at least a commutative monoid for $*$ operation). But in $\mathbb{R}$, there is no $0^{-1}$ element and thus string of symbols $0 \over 0$ or any string containing such substring is meaningless.

Formal rules of manipulation over equalities (actually, over any theory statements) have one important property: if you started from some string(s) that have meaning and apply only these rules, you can't arrive to a meaningless string. So, either one of your initial statements was meaningless or you have made a mistake in your manipulations.

Answer 2

Maybe this is essentially what you did: $1=\lim_{n\rightarrow+\infty} \frac{1/n}{1/n}='\frac{\lim_n 1/n}{\lim_n 1/n}=''\frac00$, hence $\frac00=1$. But that $='$ was not true (not an equality), as $\lim_n \frac{a_n}{b_n}=\frac{\lim_n a_n}{\lim_n b_n}$ has been proved for $\lim_n b_n\ne0$ only and is generally false when $\lim_n b_n=0$.

More generally, there is no way of proving $a=\frac00$ or $\frac00=a$ for any real or complex number $a$ or extended real number $a\in[-\infty,\infty]$, because that is not true. So to obtain that through manipulation, you must use some nonallowed manipulation, like $a=b \Rightarrow a=b+1$ or the above manipulation.

As said above, $\frac00$ is meaningless (as we are talking about the real number $0$ or the like). It has not been defined, because no reasonable definition exists, as $x\cdot0=0$ for any real number $x$. Such whimsy definitions (e.g., $x^0=1$) are made only when they allow us to neatly shorten our notation (and usually hence have some kind of "logic explanation" too).

But even though $='$ is not true, one could ask whether the $=''$ is true, or almost equivalently, whether $\frac00=\frac00$. You are not allowed to even write "$\frac00$" if your universe is the set of (extended) real or complex numbers or the like. But if you are talking about elements in a strange set containing the symbol $\frac00$ (which could be short-hand for something), then, of course, that element equals itself, i.e., $\frac00=\frac00$. But then that notation need not have anything to do with zeros or their divisions.

Although this and the above comments may make your question seem "silly", it is not stupid. I guess that most mathematicians wonder such things at some point of development, then later they get more used to logic and see such questions as trivial, like "how could I ever have wondered such a trivial thing?" So you may have a great future.