I understand how this satisfies the definition typically used for ordered pairs. But how does this satisfy the requirement that every element $x$ in $A$ must be a subset of set $A$? I know that $A$ can also be written as $\{\{\{a\}, \{a, b\}\}, \{\{b\}, \{b, c\}\}, \{\{a\}, \{a, c\}\}\}$, but fail to see how every element of this is subset of $A$, and furthermore how $\{\{\{a\}, \{a, b\}\}, \{\{b\}, \{b, c\}\}, \{\{c\}, \{c, a\}\}\}$, for instance, differs in any substantial way. Thanks for any help with this.
Asked
Active
Viewed 75 times
0
-
4They aren't "alternative definitions". They are definitions of different concepts that use the same word. Sets are "transitive" when the relation "is an element of" is a transitive relation on a particular set, so that the set $A$ is transitive if and only if for all $x,y$, if $x\in y$ and $y\in A$, then $x\in A$. This is a different concept than the concept of an equivalence relation on a set $X$ being transitive (though the name is derived from the same idea, as noted above). Your set $A$ defines a transitive relation on ${a,b,c}$, but is not a transitive set. – Arturo Magidin Sep 26 '22 at 02:41
-
Ah, that makes a lot more sense. I was very confused on this, thanks. – Epic Cabbage Sep 26 '22 at 02:44
-
1This occurs several times throughout mathematics. The word "Normal" for instance appears no fewer than $20$ times to be used to describe entirely different concepts. – JMoravitz Sep 26 '22 at 12:37