Set theory, equality of sets

Question

Proove or disprove by example

If $\{x_1,x_2\}=\{y_1,y_2\}$ then (12) or (21) holds.
(12) $x_1=y_1$ and $x_2=y_2$; (21) $x_1=y_2$ and $x_2$=$y_1$.

I am mistaking something or could I just set $x_1=x_2=y_1$ and $y_2=\emptyset$ and disprove the whole thing?

Moreover, couldn't I use the same trick again for this?
If $\{x_1,x_2,x_3\}= \{y_1,y_2,y_3\}$ then at least one of the six (123)-(321) holds.
(123) $x_1=y_1$, $x_2=y_2$, $x_3=y_3$.
(132) $x_1=y_1$, $x_2=y_3$, $x_3=y_2$.
(213) $x_1=y_2$, $x_2=y_1$, $x_3=y_3$.
(231) $x_1=y_2$, $x_2=y_3$, $x_3=y_1$.
(312) $x_1=y_3$, $x_2=y_1$, $x_3=y_2$.
(321) $x_1=y_3$, $x_2=y_2$, $x_3=y_1$.

score 2 · Accepted Answer · edited Apr 13 '17 at 12:21

2

This is a common error: the emptyset isn't nothing! For example, $\{\emptyset\}$ and $\{\}$ are very different things (think about the difference between an empty bag, and a bag containing an empty bag). So, in particular, if we let $x_1=x_2=y_1=a$ for some $a\not=\emptyset$, then $\{x_1, x_2\}=\{a\}$, but $\{y_1, y_2\}=\{a, \emptyset\}\not=\{a\}$.

This issue has been treated on this site (and in a number of other places); see e.g. here.

edited Apr 13 '17 at 12:21

Community

1

answered Oct 10 '16 at 19:54

Noah Schweber

245,398

I see now, why it won't work for the first one, but in case of the 2nd one, I simply could ${1,1,2}={1,2,2}$, so the 2nd won't hold. Tyvm – Ben373 Oct 10 '16 at 20:00
@Ben373 Yes, that's correct - the second one doesn't hold. – Noah Schweber Oct 10 '16 at 20:08

Set theory, equality of sets

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