Math Help - prove whether the set property is true, false, or sometimes true and sometimes false?

1. prove whether the set property is true, false, or sometimes true and sometimes false?

Sorry, I posted a similar question earlier, but I kind of messed it up. So here's the correct one.

Q: If B is a proper subset of C, then C - B does not equal the empty set. Is this statement always true, always false, or sometimes true and sometimes false? Explain.

I get that if B is a proper subset of C, then there must be at least one element within the set C that doesn't belong in set B. And for C - B to take place, I explained that x must be a member of C and can't be a member of B. But I really don't know how to tie these relations together. Am I getting hot or cold?

2. Re: prove whether the set property is true, false, or sometimes true and sometimes fa

Originally Posted by Taurus3
If B is a proper subset of C, then C - B does not equal the empty set. Is this statement always true, always false, or sometimes true and sometimes false? Explain.
if B is a proper subset of C, then there must be at least one element within the set C that doesn't belong in set B. And for C - B to take place, I explained that x must be a member of C and can't be a member of B.
That is all correct.

3. Re: prove whether the set property is true, false, or sometimes true and sometimes fa

wait, but how do tie all these together? Like is this always true?

4. Re: prove whether the set property is true, false, or sometimes true and sometimes fa

Originally Posted by Taurus3
wait, but how do tie all these together? Like is this always true?
Tie what together?
The statement that B is a proper subset of C means
$\begin{gathered} \bullet ~B \subset C \hfill \\ \bullet ~B \ne \emptyset \hfill \\ \bullet ~B \ne C \hfill \\ \end{gathered}$

5. Re: prove whether the set property is true, false, or sometimes true and sometimes fa

It does not mean that $B\ne\emptyset$...

6. Re: prove whether the set property is true, false, or sometimes true and sometimes fa

wait, then can I still conclude that this is always true by the 1st and 3rd statement?

7. Re: prove whether the set property is true, false, or sometimes true and sometimes fa

Originally Posted by emakarov
It does not mean that $B\ne\emptyset$...
That depends upon whom you ask.
I have seen it defined both ways.

8. Re: prove whether the set property is true, false, or sometimes true and sometimes fa

Originally Posted by Taurus3
And for C - B to take place, I explained that x must be a member of C and can't be a member of B.
More precisely, for x ∈ C - B to take place, x must be a member of C and can't be a member of B. Also, C - B does not equal the empty set iff there exists an x ∈ C - B. Taken together, C - B does not equal the empty set iff there exists an x such that x is a member of C and not a member of B. Now compare this with
Originally Posted by Taurus3
B is a proper subset of C, then there must be at least one element within the set C that doesn't belong in set B.