Apologies if this is in the wrong forum, but I chose to post here as the question pertains to equivalence relations and classes.(adsbygoogle = window.adsbygoogle || []).push({});

Sorry if it's such a trivial question, but what is the mathematical difference between equivalence and equality? My understanding is the following, but I'm a little bit unsure (studying equivalence relations has caused my brain to have a bit of a meltdown):

Equality:- Two mathematical objects areequalif they are, in actual fact, two different representations of the same object. (e.g. [itex]a=b[/itex] means that [itex]a[/itex] and [itex]b[/itex] are two different labels for the same quantity)

Equivalence:- Two mathematical objects that are distinct, but share the same result under a particular operation areequivalentwith respect to that given operation. (e.g. The two ordered pairs [itex](a,b)[/itex] and [itex](c,d)[/itex] are not equal, in general, (they are distinct mathematical objects) but they are equivalent with respect to the relation [itex]ad=cb[/itex]).

This confusion has arisen for me through studying equivalence relations and equivalence classes, particular from the abstract notation [itex]a~b[/itex].

Given a particular equivalence relation, would it be correct that if [itex]a~b[/itex], then under this relation [itex]a[/itex] and [itex]b[/itex] can be treated as the same object, i.e. [itex]a\equiv b[/itex]?

When it comes to equivalence classes, if one partitions a set into equivalence classes and then wants to study a particular equivalence relation, is the point that given an equivalence class that satisfies that relation, [itex][a][/itex], one is free to choose any element from that equivalence class when using the particular equivalence relation as they lead to the same result?!

For example, from a physics perspective, we know that any two Lagrangians that differ by a total derivative lead to the same equations of motion. As such, can one define an equivalence relation [tex]\mathcal{L}\equiv\mathcal{L}+\frac{df}{dt}\;\;\iff\;\; \delta S_{1}=\delta S_{2}[/tex] where [itex]S_{1}=\int\mathcal{L}\; dt[/itex] and [itex]S_{2}=\int[\mathcal{L}+\frac{df}{dt}]\; dt[/itex].

As such, when describing the dynamics of a particular theory there is an equivalence class of Lagrangians [itex][\mathcal{L}][/itex] from which one can choose from, in which all of the Lagrangians can be treated as "the same" as they all lead to the same equations of motion.

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# Difference between equivalence and equality

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