# Please help: If $a \not\subset B$ and $a \in A$, is $a$ an element of $B$?

I'm learning set theory and this is a homework question. I answered, "not necessarily because nothing is telling us whether $$a$$ is an element of $$B$$ and an element cannot be a subset". Am I right? What is the correct answer?

• The answer is correct, the reasoning isn't (or worded weirdly). I would say that since $A$ isn't a subset, there is an element of $A$ that isn't in $B$, so hence, $a$ could be that element. Jul 25 '19 at 3:01
• I think it would be good to give a specific counterexample Jul 25 '19 at 3:01
• Try $B=\emptyset$ Jul 25 '19 at 3:05
• @freethinker36 Given that we seem to have misinterpreted your question, it seems that the set $A$ has nothing to do with the question. Does this seem right to you? Jul 25 '19 at 3:16
• I think it is a trick question. Jul 25 '19 at 3:19

Usually you should come up with a counterexample rather than using ambiguous descriptions.

Counterexample: Let $$A=\{0,1\}$$ and $$B=\{0\}$$ and $$a=1$$, then clearly $$A$$ is not a subset of $$B$$. However, $$a\in A$$ but $$a\notin B$$.

Thanks to @Sambo, if we take $$a=0$$, then $$a$$ is both in $$A$$ and $$B$$. Therefore we conclude: If $$A\not\subset B$$, then “$$a\in A$$” has nothing to do with “$$a\in B$$”.

• In your example, $a$ is an element of $B$ Jul 25 '19 at 3:03
• I would add a second case, $a=0$, to show that the statement "$x\in B$" could be either true or false. Jul 25 '19 at 3:04
• @J.W.Tanner Sorry sorry. I’ve seen it and fixed it.
– Feng
Jul 25 '19 at 3:05
• @freethinker36 The operators $\subset$ and $\not\subset$ are only defined for sets. So if $a$ is not a set, it makes no sense to say “$a\not\subset B$”.
– Feng
Jul 25 '19 at 3:29
• @freethinker36 Also, if your problem is like this, then $A$ has nothing to do with $B$. That would make this problem meaningless.
– Feng
Jul 25 '19 at 3:31

You are correct. We know that there are elements of $$A$$ which are not in $$B$$, and there may be elements of $$A$$ that are in $$B$$. If we pick an element of $$A$$, it could fall into either of these two categories.