Difference between equivalence and equality

["Don't panic!"]

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 equal if they are, in actual fact, two different representations of the same object. (e.g. $a=b$ means that $a$ and $b$ 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 $(a,b)$ and $(c,d)$ are not equal, in general, (they are distinct mathematical objects) but they are equivalent with respect to the relation $ad=cb$).

This confusion has arisen for me through studying equivalence relations and equivalence classes, particular from the abstract notation $a~b$.
Given a particular equivalence relation, would it be correct that if $a~b$, then under this relation $a$ and $b$ can be treated as the same object, i.e. $a\equiv b$?
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, $[a]$, 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 $$\mathcal{L}\equiv\mathcal{L}+\frac{df}{dt}\;\;\iff\;\; \delta S_{1}=\delta S_{2}$$ where $S_{1}=\int\mathcal{L}\; dt$ and $S_{2}=\int[\mathcal{L}+\frac{df}{dt}]\; dt$.
As such, when describing the dynamics of a particular theory there is an equivalence class of Lagrangians $[\mathcal{L}]$ 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.

 

[DEvens]

Sure. They can be treated as "the same" with respect to the feature or characteristic that makes them members of an equivalence class.

Just be careful to "round up the usual suspects" when you are dealing with this. There are probably other features for the elements of the equivalence class that make the elements distinguishable. And the equivalence classes that can be defined in a set may have relationships that are very different from the members of those sets. The usual example is vehicles, motor cycles, and submarines.

 

["Don't panic!"]

I guess I find the notation a little hard to get my head around (I find details it gets in the way of my understanding). I understand that when one states that two mathematical objects are equal, then they represent the same value or entity. However, one can also have conditional equality, for example, equations are conditional equalities in the sense that they are only true for a specific set of values (e.g. $x^{2} =2$, for $x\in\mathbb{R}$, has solution set $\lbrace - \sqrt{2},\sqrt{2} \rbrace$ for which this equality is true). My confusion arises with the notation $~$ used when defining an equivalence relation, for example when defining the set of rational numbers $\mathbb{Q}$ one gives the equivalence relation $$(a, b) ~(c, d) \iff ad=bc $$ and defines the set of rational numbers to be the set of equivalence classes formed under this equivalence relation. One can then write, for example $$\frac{1}{2} =\frac{2}{4}=\cdots$$ such that, even those these two fractions represent distinct quantities in their own right, under this equivalence relation they represent the same value, i.e. they're equal.
Is it the case that when one defines an equivalence relation and then wishes to study a particular system using that equivalence relation, then one can consider any two objects in the equivalence class formed under that equivalence relation as "the same" in the sense that we can choose anyone of the elements in our equivalence class to complete the analysis (going back to my example using fractions, if we were looking at something which required to multiply a number by 0.5, then we could equally use $\frac{1} {2}$, or $\frac{2} {4}$, or $\frac{5} {10}$, etc. as they represent the same numerical value)?!

Sorry to harp on a bit about the same thing, but I'm mainly getting confused with the notation and how equivalence relations are useful actual applications?!

 

[lavinia]

I guess I find the notation a little hard to get my head around (I find details it gets in the way of my understanding). I understand that when one states that two mathematical objects are equal, then they represent the same value or entity. However, one can also have conditional equality, for example, equations are conditional equalities in the sense that they are only true for a specific set of values (e.g. $x^{2} =2$, for $x\in\mathbb{R}$, has solution set $\lbrace - \sqrt{2},\sqrt{2} \rbrace$ for which this equality is true). My confusion arises with the notation $~$ used when defining an equivalence relation, for example when defining the set of rational numbers $\mathbb{Q}$ one gives the equivalence relation $$(a, b) ~(c, d) \iff ad=bc $$ and defines the set of rational numbers to be the set of equivalence classes formed under this equivalence relation. One can then write, for example $$\frac{1}{2} =\frac{2}{4}=\cdots$$ such that, even those these two fractions represent distinct quantities in their own right, under this equivalence relation they represent the same value, i.e. they're equal.
Is it the case that when one defines an equivalence relation and then wishes to study a particular system using that equivalence relation, then one can consider any two objects in the equivalence class formed under that equivalence relation as "the same" in the sense that we can choose anyone of the elements in our equivalence class to complete the analysis (going back to my example using fractions, if we were looking at something which required to multiply a number by 0.5, then we could equally use $\frac{1} {2}$, or $\frac{2} {4}$, or $\frac{5} {10}$, etc. as they represent the same numerical value)?!

Sorry to harp on a bit about the same thing, but I'm mainly getting confused with the notation and how equivalence relations are useful actual applications?!

An equivalence relation divides a set into disjoint subsets. To objects are called "equivalent" if they both lie in the same subset. So for instance, the integers can be divided into two disjoint subsets, the even and the odd integers. In this case, two integers are equivalent if they are both even or both odd. This is not the same as them being equal.
One could describe this equivalence by saying that two integers are equivalent if their remainder on division by two is the same.

Two elements are equal if in fact they are the same element. For instance 1/2 is equal to 2/4.
If you consider the set of all algebraic symbols that represent the same number, then this is an equivalence relation. So for instance the symbols 1/2 and 2/4 are equivalent since they represent the same number. 5/10 is another symbol in the same equivalence class.

If you like, you can always divide a set into the subsets that contain exactly one element in the set. This gives an equivalence relation and two elements are equivalent if they lie in the same subset, that is if they are equal.

 

[Stephen Tashi]

I understand that when one states that two mathematical objects are equal, then they represent the same value or entity.

That's correct, in spirit, but you are just substituting the concept of "same" for the concept of "equal". The concept of "same" isn't precise. For example 1 + 1 and 2 are "the same" as representations of integers, but they are different as two strings of symbols. "Sameness" and "equality" are "the same" concept!

All statements in mathematics about things being "equal" are actually statements about things being "equal with respect to" some equality relation. When the equality relation is obvious from the context, people just say "is equal" instead of saying "is equal with respect to the equality relation ...". For example in an elementary algebra text, you would write "$x = 5$ " instead of "$x \equiv 5$ with respect to the equivalence relation defined on the real numbers".

Is it the case that when one defines an equivalence relation and then wishes to study a particular system using that equivalence relation, then one can consider any two objects in the equivalence class formed under that equivalence relation as "the same" in the sense that we can choose anyone of the elements in our equivalence class to complete the analysis

It is often the case that you can do that and often the ability to do that is a major motivation for defining equivalence classes. However, you must consider things on a case-by-case basis. For example, if you are running a cryptographic program and it encodes "32/64" to a string, it doesn't necessarily encode "1/2" to the same string.

Sorry to harp on a bit about the same thing, but I'm mainly getting confused with the notation and how equivalence relations are useful actual applications?!

Your previous thought about using one symbol as the "representative" for an equivalence class is the correct motivation for applying equivalence relations. There are often situations where two things are not identical, but are interchangeable for certain purposes. Common examples in mathematics are the various definitions of "isomorphic". The relation of "isomorphism" defines an equivalence relation. The definition of a particular kind of isomorphism also contains the specifications of which particular expressions can be computed by using representatives of the equivalence classes it defines.

 

["Don't panic!"]

All statements in mathematics about things being "equal" are actually statements about things being "equal with respect to" some equality relation

So essentially, when we state that two mathematical objects are equal, in general, this is a relative statement as it won't necessarily be true that they are always equal, unless they are identical. So, under certain constraints/conditions the two objects are equal if they have the same value (or two different expressions are equal if they represent the same mathematical object). If two objects (or expressions) are identical, then this is an equality that is always true (i.e. it holds under all conditions/ values of appropriate variables). Is this correct?

 

[Stephen Tashi]

So essentially, when we state that two mathematical objects are equal, in general, this is a relative statement

Yes.

as it won't necessarily be true that they are always equal

If we say two thing are equal in one context, it is understood to mean they are equivalent with respect to some equivalence relation. They may not be equal in another context, where "equal" refers to a different equivalence relation.

, unless they are identical.

I don't know how you can define "identical" except as "equal with respect to ..." some equivalence relation.

We could try saying : " 'A is identical to B' is defined to mean that for each equivalence relation R, A is equivalent to B with respect to R."

There might be something inherently paradoxical about that definition. Perhaps a logician can tell us.

So, under certain constraints/conditions the two objects are equal if they have the same value (or two different expressions are equal if they represent the same mathematical object). If two objects (or expressions) are identical, then this is an equality that is always true (i.e. it holds under all conditions/ values of appropriate variables). Is this correct?

No. Your wording is too imprecise. You are trying to talk about mathematical equivalence using phrases like "have the same value" or "represent the same mathematical object", as if the relation of "sameness" already has a mathematical definition. To be precise you'd have to define what you mean by "have the same value" or "represent the same object".

 

["Don't panic!"]

No. Your wording is too imprecise. You are trying to talk about mathematical equivalence using phrases like "have the same value" or "represent the same mathematical object", as if the relation of "sameness" already has a mathematical definition. To be precise you'd have to define what you mean by "have the same value" or "represent the same object".

Sorry, I couldn't really think of a way to articulate it very well. Could you provide me with a better (more mathematical description /explanation)?
It would be much appreciated :-)

 

[Stephen Tashi]

Sorry, I couldn't really think of a way to articulate it very well. Could you provide me with a better (more mathematical description /explanation)?

I'm not sure what you are trying to express.

If two objects (or expressions) are identical, then this is an equality that is always true (i.e. it holds under all conditions/ values of appropriate variables). Is this correct?

That refers to an equation that expresses an "identity" (like x + x = 2x). Such an identity involves a specific equivalence relation. In elementary algebra the "=" refers to the equivalence relation defined for real and complex numbers. An identity is a true statement for all values of its variables.

There can be mathematical identities involving things other than numbers. For example, in set theory, for any set $A$ we have $A \cup A = A$. The "=" in that identity refers to the equality relation defined on sets, which is different than the equality relation defined on numbers.

 

["Don't panic!"]

I'm not sure what you are trying to express.

I guess a mathematical definition of equality. Of course I have the intuitive notion, but this is, as you say, imprecise as it uses words like "same". On Wikipedia it gives this sort of heuristic definition also: "In mathematics equality is a relationship between two quantities or, more generally two mathematical expressions, asserting that the quantities have the same value or that the expressions represent the same mathematical object." but this seems unsatisfactory as it is just as imprecise as the explanation I gave?!