Equivalence Relations Reasoning

Question

Equivalence relations are in terms of a single set and described as follows:

1) any element being related to itself,

2) symmetry of elements' relationships; $\space$an element related to another implies the latter element is also related to the former,

3) transitivity of elements' relationships;$\space$ for any element related to another and that other element related to yet another, the first element is also related to the last (and, further, symmetry implies each of the three is related to the other two).

This is all fine and dandy if you want to say that some elements of a set are functionally equivalent to other elements of that same set, but I think that's not the point of equivalence relations. So what if you want to consider two different sets? Particularly, what if I want to do basic mathematics with dots instead of numbers, each dot representing the number 1, then how do I say the set {1 dot, 2 dots, 3 dots, ...} is equivalent to the naturals? Must I union the dots set and the naturals and manually construct the relation set with whatever (i, j) relationship over the naturals, i and j being natural numbers, implying the analogous relationships (i dots, j), (i, j dots), and (i dots, j dots)? Is this the same as manually constructing a bijective linear function between {1 dot, 2 dots, 3 dots, ...} and the naturals?

How is this generalized to everyday life? Say if you have a fleeting idea and need to make a physical note of it immediately, then a pencil is equivalent to a pen and either will suffice. Or if you need to make a call and already know the number to call, then any phone is equivalent to any other; your cell phone, a friend's cell phone, or an office phone are all able to make the call. Or if you're cleaning your apartment's floor with floor cleaning solutions, then any two floor cleaning solutions of different brands but the same chemical composition are equivalent and one will clean just as well as the other.

Answer 1

how do I say the set {1 dot, 2 dots, 3 dots, ...} is equivalent to the naturals?

You're right that this is a different type of equivalence. What you're looking for is often called an "isomorphism" in various technical contexts. As you guessed, it's a specific function from one set to another - a transformation of natural numbers into dots - in such a way that you don't lose whatever structure you want to work with (e.g. addition, subtraction and multiplication can either be done on the numbers or on the dots, and both will give you the same answer under this transformation function).

There are more general notions too. An "equivalence of categories" is a more general notion: even when there doesn't exist a precise one-to-one mapping between two collections of things, you can often say that they work in equivalent ways.

I won't go into detail, because the details are highly technical. But yes, these notions exist.

How is this generalized to everyday life?

I don't think I really understand this question. If anything, perhaps it is a concept that has been borrowed from everyday life. But you've got the idea right: two things are somehow functionally equivalent if they can be used in the same way, in the same context, for the same purpose. A pen and a pencil are equivalent in a generic context where you need to write something. There are also situations where they're not equivalent: when you're signing a cheque, or (to give a silly example) when you want to try out your shiny new pencil sharpener.

In each of these contexts, rather than being overwhelmed by the many hundreds of billions of possible writing implements you could possibly choose from, your brain pares down the space of possible choices by telling you "most of these things are basically the same here". That's exactly the idea we're trying to formalise with equivalence relations.

Answer 2

In way "equivalence" is an inappropriate word.

To things are equivalent if ... we can replace something with another and for all practical purposes they act the same. An equivalence relation a way of technically saying these to things and interchangeable. Obviously something is interchangeable with itself (reflexivity!); of course you can interchange two things it goes both ways (symmetry!); and if you have several interchangeable things the must each pair be interchangeable (transitivity!).

Thus the "equivalence relation" we all love.

This actually a very high level concept although it feels very low level. We you want to say Dots are equivalent to Numbers the "things" that are equivalent (or interchangeable) are not the individual dots with individual numbers but that the "things" are sets. $\mathbb N$ is one THING (not a set of things but a thing). And $Dots =\{some set of dots somehow distinguished\}$ is another THING. And these two THINGS are equivalent in some equivalence relation between SETS.

So let $U = \{sets\}$ and we say if $A \in U$ is a set and $B\in U$ is a set and $R$ is a relation between sets so that .... um.... so that what, what exactly is the relation between $A$ and $B$?

$R$ is the condition that, $a R b$ if $a$ and $b$ are sets and there exists a function $h: a \to b$ so that ... they keep basic structure.

Okay, I've been beating around the bush.

$\mathbb N \in U$ and $\mathbb N$ is a THING that has the following properties: There is an operation called $+$ and an operation called $\times$ and they are closed binary operations on $\mathbb N$.

So $a R b$ if $a$ also have operations called $+_a$ and $\times_a$ and $b$ has $+_b$ and $\times_b$ and there is a function $h: a \to b$ so that for any two $x,y \in a$ then $h(x+_a y) = h(x)+_b h(y)$ and $h(x\times_a y) = h(x)\times_b h(y)$.

[Notice this means if $a$ has (or doesn't have) any axiom like commutativity, or associativity or distribution or identity elements or inverse elements than if $a R b$ then $b$ must have the exact same properties.]

So $Dots R \mathbb N$ if there are a function $h:\mathbb N \to Dots$ so that $h(a +|\times b) = h(a) +_{dots}|\times_{dots} h(b)$

[Note: and there must but a function $g:Dots\to \mathbb N$. ANd that $g(h(n+|\times m))= n+|\times m$. Thus $h$ must be a bijection. ]