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Apologies if this is in the wrong forum, but I chose to post here as the question pertains to equivalence relations and classes.

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 are

Equivalence:- Two mathematical objects that are distinct, but share the same result under a particular operation are

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.

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 are

*equal*if 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 are

*equivalent*with 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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