If A ⊂ B, then A ∩ B = A ∪ B. Always, Sometimes, or Never.

Question

If A ⊂ B, then A ∩ B = A ∪ B.  Always, Sometimes, or Never.

zzr0ck3r · Answer

sometimes

zzr0ck3r · Answer

if $A=B$ then for sure this is true
for counter examples

$\mathbb{N}\subset \mathbb{Z}$ but $\mathbb{N}\cap\mathbb{Z}=\mathbb{N}\ne\mathbb{N}\cup\mathbb{Z}=\mathbb{Z}$

anonymous · Answer

But $A$ cannot be $B$ since the former is a proper subset of the latter.

zzr0ck3r · Answer

$\subset$ often just means subset

zzr0ck3r · Answer

I dont even know how to do the other one in latex

anonymous · Answer

Yes, but your answer is for $\subseteq $

zzr0ck3r · Answer

lol im saying many people use these two things to mean interchangable.

zzr0ck3r · Answer

in my analysis book they mean the same thing, and my abstract algebra book they are different

anonymous · Answer

$\subset$ and $\subseteq $ are different the same way $< $ and $\le $ are.

zzr0ck3r · Answer

not always....

|dw:1403590266757:dw|

zzr0ck3r · Answer

some people use this...

anonymous · Answer

I mean I'm sure the question doesn't mean $\subseteq$ in this case, but a proper subset..

zzr0ck3r · Answer

the notation is used differently in different places and it is not universal

anonymous · Answer

sigh

zzr0ck3r · Answer

yeah dude... I get your point you dont get mine..

anonymous · Answer

well i got marked incorrect for sometimes...

anonymous · Answer

^ lel

zzr0ck3r · Answer

then its inclusive.

anonymous · Answer

rip.

too late.

zzr0ck3r · Answer

dude @zeta i can show you many books where they mean the exact same, again it is not universal

zzr0ck3r · Answer

if the user knows which one they are talking about then the user should not put the answer I used.

zzr0ck3r · Answer

for it would be obvious that what I am saying cant be true because then $A=B$ does not make sense...

anonymous · Answer

can you give an example of interchangeable use?

kainui · Answer

@Zeta Just because you possibly happen to have the same convention of symbolizing things doesn't really make you right -- it just means the person asking the question didn't give us enough context.

anonymous · Answer

zz0ck3r, there is a "never" choice too lol

zzr0ck3r · Answer

sure

intro to abstract algebra by fraleigh

he uses $\subset$ to mean general subset and |dw:1403590599163:dw|
to mean proper.

I can list 20 more books...