Testing whether "$\emptyset$ in $ \emptyset$ " is an equivalence relation

Question

I'm doing exercise 4.1(e):

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The answer key says that the relation above is an equivalence relation. I don't even know what the relation they've defined means.

This is how I'm interpreting: Let $ \in$ define the relation. I believe "in" means $\in$.

Pf: Reflexive $\iff$ Symmetric

$\emptyset$ $ \in$ $\emptyset$

Transitive: if $\emptyset$ $ \in$ $\emptyset$ and $\emptyset$ $ \in$ $\emptyset$, then $\emptyset$ $\in$ $\emptyset$ .

therefore $ \in$ is an equivalence relation?

Endpf.

Explanation requested.

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Someone of doubt also claims that

" $ \emptyset$ in A, where A is a non-empty set"

is an equivalence relation?. I have no idea what "in" means.

Answer 1

This is indeed somewhat confusing; let me break down what they're talking about.

The source of your confusion is I think the use of the word "in." In this context, it does not mean "$\in$." When Hrbacek/Jech say "Is $E$ a [someting] relation in $X$?," it might be better for them to use the word "on" - what they mean is that $X$ is the underlying set, and $E$ is the relation. I'll be using their meaning of the word "relation in $X$" throughout this answer, even though I disapprove.

Now what about the specific problem(s) here? Well, these exercises are designed to do two things. The first is to show you that the choice of the underlying set can matter: e.g. the relation $E=\{(a, a)\}$ is reflexive in $\{a\}$ (since every element of $\{a\}$ is related to itself) but not in $\{a, b\}$ (since $b$ is not related to itself: $(b, b)\not\in E$).

The second point is to give you some practice reasoning in a nonintuitive context - specifically, about the empty set. $\emptyset$, as usual, is a confusing object. Remember that "for all" statements about the empty set are vacuously true - for instance, $\emptyset$ is a set of elephants, since every element of $\emptyset$ is an elephant. More relevantly here, $\emptyset$ is a set of ordered pairs, hence a relation (it's the relation which never holds).

Now as long as our domain isn't stupid, $\emptyset$ clearly won't be reflexive: no element will be related to itself, since no element will be related to anything (our relation never holds). But what if we take our underlying set to also be $\emptyset$? Then every element of $\emptyset$ is related to itself - remember that "for all" statements about $\emptyset$ are vacuously true!

So $\emptyset$ is a reflexive relation if our domain is $\emptyset$. And it's clearly transitive and symmetric no matter what the domain is (exercise). So, to use the phrasing of the book, $\emptyset$ is an equivalence relation in $\emptyset$.

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Re: the end of your question, you say that

The book also claims that $\emptyset$ is an equivalence relation in a nonempty set $A$.

This is false - $\emptyset$ isn't reflexive in any nonempty set $A$ (why?). Where does the book claim this? ${}{}$

Answer 2

Yes $\emptyset$ an equivalence relation on $\emptyset$. What is $\emptyset$ as a binary relation? It's the relation such that no two elements are related.

However, if $A$ is not empty, then $\emptyset$ is not an equivalence relation on $A$ (and the authors do not claim that it is), because if $a\in A$, then it is not true that $(a,a)\in\emptyset$.

Answer 3

A relation $R$ is said in a set $A$ when the $\operatorname{field} R\subseteq A$, where the $\operatorname{field} R=\operatorname{dom}R\cup\operatorname{range}R$. It is also said in this case that $R$ is a relation between elements of $A$.

Answer 4

The precise definition of a relation is that it is a set of ordered pairs. We then say that x~y if (x,y) is in that set. However, that's a rather cumbersome definition; if one wishes to discuss the relation of one integer being larger than another, it would not be possible to list every ordered pair in which the first is larger than the second. So, in practice we often dispense with the set-theoretic definition and instead define relations by giving a proposition that has two free variables. For instance, "x is the square root of y" would be a proposition with two free variables: x and y. The relation is then taken to be the set of ordered pairs for which the proposition is true. So in this case (0,0), (1,1), and (2,4) would be in the relation, but (3,6) would not.

What's confusing about this list (besides the use of "in" rather than "on") is that the first four relations are given in this propositional form, while the last two are in the set form. The fifth one has no free variables, and the sixth has only one, so asking whether they are equivalence relations is incoherent in the context of using the propositional definition of relations.

P.S. it's good practice when presenting an image to also give a transcript of what's in the image.