I'm getting a little confused with sets and subsets.
Which of the following is a member of {x,y,z}?
"x" or {x}?
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x3nr0s
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{x} is a member of {{x},{y},{z}} – randomgirl Feb 10 '15 at 21:16
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$x$ is. ${x}$ is not (it is a subset of ${x,y,z}$ (relation $\subseteq$), not an element (relation $\in$)). – Clement C. Feb 10 '15 at 21:17
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Informally, $\alpha$ is a member of a set $A={\cdots}$ iff $\alpha$ is somewhere written exactly as it is in $\cdots$. – user26486 Feb 10 '15 at 21:17
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Also {x} is a subset of {x,y,z} – randomgirl Feb 10 '15 at 21:18
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The dual question: http://math.stackexchange.com/questions/491465/is-emptyset-a-subset-of-emptyset/491468#491468 – Asaf Karagila Feb 10 '15 at 21:20
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Thanks. Would that mean that {x} is not member of {{x,y}}? – x3nr0s Feb 10 '15 at 21:20
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Indeed, it is not. – Clement C. Feb 10 '15 at 21:21
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$x$ is a member of the set, $(x)$ is it self a set so it cannot be a member of your set $(x,y,z)$ unless your sets is comprises $((x),(y),(z))$. – Hedwig Feb 10 '15 at 21:22
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see here – user 1 Feb 11 '15 at 06:56
1 Answers
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Given the set $\{x,y,z\},$ we say that $x$ is a member of the set.
It is also true that $\{x\}$ is a subset of the set $\{x,y,z\}.$
Addendum: A set can be a member of another set. For example, $\{x\}$ is a member of the set $\{\{x\},y,z\}.$ But in that case, observe the extra $\{\cdot\}$ brackets around $x$ that do not appear in the notation for the set $\{x,y,z\}.$ The sets $\{\{x\},y,z\}$ and $\{x,y,z\}$ are two quite different sets.
David K
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1@Xenidious It is not a subset, since $x$ is not a member of ${{x},{y},{z}}$, i.e. it is not the case that all the members of ${x}$ are in ${{x},{y},{z}}$. By definition, $A\subseteq B\iff (x\in A\implies x\in B)$. – user26486 Feb 11 '15 at 00:15