0
$\begingroup$

I just learnt the difference between Inclusion ($ \subset $) and Belonging ($ \in $). I also learnt as an example,

for the two sets $A=\{4\}$ and $B=\{4, 5\}$, $A \not\subset B$, but $A \subset \{\{4\},5\}$. -- (I)

However, Halmos's 'Naive Set Theory' book, as well as few posts like this and this say something like

"It ($ \varnothing \subset A$) could be false only if $ \varnothing $ had an element that did not belong to A."

Wondering if my understanding of (I) is wrong! Otherwise, I can see that even if A has no element that does not belong to B, A is not a subset of B.

$\endgroup$
  • 3
    $\begingroup$ (I) is false. $A \subset B$ and $A \in \{\{4\},5\}$. By definition, for two sets $A,B$ we have $A \subseteq B$ precisely if every element that is contained in $A$ is also contained in $B$ $\endgroup$ – G. Chiusole Aug 20 '19 at 14:58
  • 1
    $\begingroup$ Maybe have a look at this question $\endgroup$ – G. Chiusole Aug 20 '19 at 15:04
1
$\begingroup$

Your (I) is indeed false. "$A\subseteq B$" means "Every element of $A$ is an element of $B$," and it should be clear that every element of $A$ (there's only one - $4$) is also an element of $B$.

I've used "$\subseteq$" instead of "$\subset$" since some texts use the latter to refer to proper subsets. Also, it's worth noting that the inclusion/belonging terminology isn't universal mathematically, and in particular "$A$ is included in $B$" could reasonably be misunderstood as "$A\in B$."

Now if you replace your $A$ with "$\{\{4\}\}$," then the statement is indeed true, and it's possible (likely, even) that this is what was intended and that there was a typo or mishear/misreading issue.

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ @ Noah Schweber - You are very generous with your last para, but it was more of wrong understanding from my part. Thanks to you and 'G. Chiusole'. $\endgroup$ – KGhatak Aug 20 '19 at 15:23

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.