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

• Oct 4th 2011, 08:58 AM
Taurus3
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?
• Oct 4th 2011, 09:07 AM
Plato
Re: prove whether the set property is true, false, or sometimes true and sometimes fa
Quote:

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.
• Oct 4th 2011, 10:17 AM
Taurus3
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?
• Oct 4th 2011, 10:24 AM
Plato
Re: prove whether the set property is true, false, or sometimes true and sometimes fa
Quote:

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
$\displaystyle \begin{gathered} \bullet ~B \subset C \hfill \\ \bullet ~B \ne \emptyset \hfill \\ \bullet ~B \ne C \hfill \\ \end{gathered}$
• Oct 4th 2011, 01:00 PM
emakarov
Re: prove whether the set property is true, false, or sometimes true and sometimes fa
It does not mean that $\displaystyle B\ne\emptyset$...
• Oct 4th 2011, 04:51 PM
Taurus3
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?
• Oct 4th 2011, 04:58 PM
Plato
Re: prove whether the set property is true, false, or sometimes true and sometimes fa
Quote:

Originally Posted by emakarov
It does not mean that $\displaystyle B\ne\emptyset$...

That depends upon whom you ask.
I have seen it defined both ways.
• Oct 5th 2011, 12:45 PM
emakarov
Re: prove whether the set property is true, false, or sometimes true and sometimes fa
Quote:

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
Quote:

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.