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Set Theory Exercises
04-04-2017, 10:23 AM
Post: #1
Set Theory Exercises
1.1 Let $(a,b):=\{\{a\},\{a,b\}\}$, then
$\quad(a,b)=(c,d)\iff (a=c)\wedge(b=d)$
Proof. Assume $(a,b)=(c,d).$ If $a\ne b$, then $c\ne d$.
$\quad$Otherwise $(c,d)=\{\{c\}\}$ and $\{a,b\}\not\in(c,d)$ so
$\quad a = c\,$(otherwise $\{a\}=\{c,d\}\implies c=d$)
$\quad$Now $(a=c)\wedge(\{a,b\}=\{c,d\})\implies (b=d)$
$\quad$If $a=b$ then ${\small((a,b)=(c,d))}\implies a=b=c=d.$
$\quad (a=c)\wedge(b=d)\implies (a,b)=(c,d)$ is trivial.
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