I'm reading Devlin, Fundamentals of contemporary set theory, I am at page 13:
Definition Let $R$ denote any binary relation on a set $x$. We say:
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$R$ is reflexive if $(\forall a\in x) (aRa)$
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$R$ is symmetric if $(\forall a,b\in x) (aRb\rightarrow bRa)$
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$R$ is antisymmetric if $(\forall a,b\in x) [(aRb \wedge a\neq b)\rightarrow (\neg bRa)]$
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$R$ is connected if $(\forall a,b\in x)[(a\neq b)\rightarrow (aRb \vee bRa)]$
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$R$ is transitive if $(\forall a,b,c\in x)[(aRb \wedge bRc)\rightarrow (aRc)]$.
Excercise: Which of the above properties does the membership relation, $\in$, on a set satisfy?
My answer is NONE. But I'm very confused by this question, since $\in$ relates objects of different nature, i.e. elements and sets of elements, so I actually don't know how to deal with this particular relation. Can you help me clarifying what is going on? Consider that this is the first chapter of the book, hence set theory is not developed in an axiomatic way at this stage, and I think the exercise should be done with no use of axiomatizations, only using intuition or evidence.
counterexamples:
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reflexivity: an element is not a member of itself
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symmetry: take $a$ an element, $b=\{a,c\}$
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antisymmetry: take $a=\{b,c\}$ and $b=\{a,d\}$, with $c\neq d$
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connection: $a$ and $b$ different elements
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transitivity: take $a$ an element,$b=\{a,d\}$ and $c=\{b,e\}$
Best Answer
I have interpreted the exercise in the following way: We are working in ZFC. In particular, this means that any element of a set is also a set. I have looked on the relation $\in$ for some simple sets which can be obtained from the empty set $\emptyset$.
When I look at the table of contents of Devlin's Joy of Sets (Fundamentals of Contemporary Set Theory), I saw that the author deals with axiomatic set theory only in the second chapter of this book. So it is possible that I have somehow misinterpreted what he wanted in this exercise.
Let us start by two trivial observations. If $x=\emptyset$, then all properties are vacuously true. If the set $x$ has one element, then $R$ is connected and antisymmetric, since we never have $a\ne b$.
The relation $\in$ on any set $x$ must be antisymmetric, since $a\in b\in a$ contradicts Axiom of regularity.
If $x=\{\emptyset,\{\emptyset\}\}$ then the relation $\in$ is not symmetric, just take $a=\emptyset$ and $b=\{\emptyset\}$. Notice that relation $\in$ on this set is connected and it is also transitive.
Let $x=\{\emptyset,\{\{\emptyset\}\}\}$. If we denote $a=\emptyset$ and $b=\{\{\emptyset\}\}$, then neither $a\in b$, nor $b\in a$. So the relation $\in$ on this set is not connected. However, as it is the empty relation, it is transitive.
Let $x=\{a,b,c\}$ where $a=\emptyset$, $b=\{\emptyset\}$ and $c=\{\{\emptyset\}\}$. Then $a\in b$ and $b\in c$, but $a\notin c$, so the relation $\in$ on this set is not transitive. It is also not connected, since neither $a\in c$ nor $c\in a$.