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?

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

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**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.

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?

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

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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} $

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

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

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?

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

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**emakarov** It does not mean that $\displaystyle B\ne\emptyset$...

That depends upon whom you ask.

I have seen it defined both ways.

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.