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What does it mean to be equal? $x$ is equal $y$, written symbolically $x=y$. I know that $=$ is a equivalence relation. I have never seen a formal definition of the relation $=$.

Sorry
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    That depends on what sort of object your are considering and how that object itself is defined. For sets, which for some ways of constructing mathematics everything is technically defined as under the hood, equality between sets is defined as $A=B\iff A\subseteq B$ and $B\subseteq A$. As we grow in complexity, we often gloss over the fact that it is sets involved, using instead the version of equality for smaller objects in the definition for equality in larger objects, for example equality for rationals $\frac{a}{b}=\frac{c}{d}$ is defined as $ad = bc$ using equality for integers – JMoravitz Nov 08 '19 at 18:02
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    @JMoravitz I think it could be an answer as well :) – Botond Nov 08 '19 at 18:03
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    It depends of what kind of definitions to you have. For example, if $A$ is a non-empty set with an order $\leq$ defined, so you have $a=b$ if $a\leq b$ and $b \leq a$ – Mateus Rocha Nov 08 '19 at 18:03
  • @JMoravitz I see your point. – Sorry Nov 08 '19 at 18:06
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    Related: https://math.stackexchange.com/questions/3034178/the-definition-of-the-word-equation-in-math/3034192#3034192. Do read http://www.math.harvard.edu/~mazur/preprints/when_is_one.pdf – Ethan Bolker Nov 08 '19 at 18:07
  • @EthanBolker the paper you link to is a delightful read. I'm rather enjoying it, and it seems to be hitting the nail on the head. – JMoravitz Nov 08 '19 at 18:25

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The first time you formally encounter $=$ will probably be in set theory, where it is a relation on sets defined as $A=B$ iff all members of $A$ are equal to all members of $B$. (And of course, the empty set equals itself).

More generally, the equality operator needs to be defined on members of sets in order to match the goals of the work you intend on doing. An equality operator on set $X$ is a binary operator, i.e., it is a function $f:X\times X\to \{=,\neq\}$, where any member $(x_1,x_2)\in f^{-1}(=)$ has the property that $x_1=x_2$. Mathematicians have introduced three criteria for a consistent notion of equality:

  1. Reflexive: $x=x$ for all $x\in X$.
  2. Symmetric: If $x=y$, then $y=x$.
  3. Transitive: If $x=y$ and $y=z$, then $x=z$.
Rushabh Mehta
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  • Alright, that makes sense. – Sorry Nov 08 '19 at 18:06
  • It is relevant to mention that these three criteria are not sufficient to define equality, the relation of similarity in triangles also satisfies those criteria – WW1 Nov 08 '19 at 18:09
  • @WW1 I don't see a problem with that. That is a perfectly valid notion of equality. – Rushabh Mehta Nov 08 '19 at 18:10
  • Perhaps I am thinking of the use of the symbol "=" rather than the relation of equality. For instance, it is interesting to ponder the meaning of "=" in the statement $$\lim_{x\to 0}\frac1{x^2}=\infty$$ – WW1 Nov 08 '19 at 18:20
  • @WW1 In complete spaces, it is possible to make an equivalence relation out of limits. – Rushabh Mehta Nov 08 '19 at 20:02