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I am reading "The Joy of Sets: Fundamentals of Contemporary Set Theory". A binary relation $R$ on a set $x$ is defined as a subset of the product $x\times x$. The well-known properties of a binary relation are then recalled:

$R$ is reflexive if $(a,a)\in R$ for every $a\in x$

$R$ is symmetric if $(a,b)\in R$ implies $(b,a)\in R$

$R$ is antisymmetric if $(a,b)\in R$ and $(b,a)\in R\to a=b$

$R$ is connected if $a=b$ or $(a,b)\in R$ or $(b,a)\in R$

$R$ is transitive if $(a,b)\in R$ and $(b,c)\in R\to (a,c)\in R$

Then Exercise 1.5.1 askes: which of the above properties are satisfied by the membership relation $\in $ on a set $x$?

I can't understand how the membership relation $\in$ is defined. I guess that it must be a subset of $x\times x$. Then $(a,b)\in \epsilon$ iff $a\in b$? So the elements of $x$ are supposed to be sets themselves?

bateman
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    "the elements of [the set] $x$ are supposed to be sets themselves?" YES; in usual set theories, every object is a set. – Mauro ALLEGRANZA Jan 15 '18 at 16:00
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    Related: https://math.stackexchange.com/questions/599851/properties-of-membership-relation-in-naive-set-theory – Air Conditioner Jan 15 '18 at 16:02
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    And yes; if we call it $\text {in}_X$, then we have: $\text {in}_X = { (a,b) \mid a,b \in X \land a \in b }$. Obviously, $\text {in}_X \subseteq X \times X$. – Mauro ALLEGRANZA Jan 15 '18 at 16:21

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