set theory notation $\in \uparrow$ in set theory

Question

I'm reading the following set of notes

http://ozark.hendrix.edu/~yorgey/settheory/index.html

and on page 6 of the full set of notes (or first page of the second link), the symbol $\in \uparrow$ is used, though $\uparrow$ isn't quite right because its more like half that....

Anyway, I don't know what it is. I think it's roughly something like the analogy of $\le$ is to $<$ as $\in$ is to $\in \uparrow$ but I'm not sure.

Any help appreciated, thank you

Answer 1

The symbol in question is $\upharpoonright$, which is used for the restriction of a function or a relation to a subset of its domain. In particular:

• If $f : A \to B$ is a function and $U \subseteq A$, then $f \upharpoonright U : U \to B$ is the function defined by $(f \upharpoonright U)(a)=f(a)$ for all $a \in U$;
• If $R$ is a relation on a set $X$ and $U \subseteq X$, then $R \upharpoonright U$ is the relation on $U$ defined for all $x,y \in U$ by $x\, (R \upharpoonright U)\, y$ if and only if $x\,R\,y$.

In this case, ${\in} \upharpoonright x$ is the restriction of the set membership relation to $x$. So what it means to say that $\langle x, {\in} \upharpoonright x \rangle$ is a well-ordering is that $x$ is totally ordered by $\in$ and every inhabited subset of $x$ has a minimal element with respect to the relation $\in$.

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P.S. the $\LaTeX$ code for $\upharpoonright$ is \upharpoonright (or \restriction — thanks Misha).