Question about sets

**Nombredor** · December 2nd 2011, 05:10 PM

If I have ((A ∩ B) ∪ Ω), is that equivalent to Ω ?

**Plato** · December 2nd 2011, 05:19 PM

Originally Posted by Nombredor

If I have ((A ∩ B) ∪ Ω), is that equivalent to Ω ?

Well of course, if $X\subset Y$ then $X\cup Y=Y~.$
All set are subsets of $\Omega$ .

**Nombredor** · December 2nd 2011, 05:29 PM

And I am right in saying that (A -B) ∪ (B - A) is equivalent to (A ∪ B) - (A ∩ B) ?

And is it accurate to say that (A ∩ Ω) ∪ ~B is the same as Ω - B ?

**Plato** · December 2nd 2011, 05:42 PM

Originally Posted by Nombredor

And I am right in saying that (A -B) ∪ (B - A) is equivalent to (A ∪ B) - (A ∩ B) ?

That is correct.

Originally Posted by Nombredor

And And is it accurate to say that (A ∩ Ω) ∪ ~B is the same as Ω - B ?

If ~B means $B^c$ , the complement of $B$ then no.
$(A\cap\Omega )\cup B^c=A\cup B^c$ which is not necessarily $B^c~.$

**Nombredor** · December 2nd 2011, 05:45 PM

~B means ''non B'' in this case.

**Plato** · December 2nd 2011, 05:49 PM

Originally Posted by Nombredor

~B means ''non B'' in this case.

"non B" is not a set theory term.
B complement is the set of all elements not in B.

**Nombredor** · December 2nd 2011, 05:52 PM

Originally Posted by Plato

$(A\cap\Omega )\cup B^c=A\cup B^c$ which is not necessarily $B^c~.$

How do they look different on a Venn diagram ?

**Nombredor** · December 2nd 2011, 06:14 PM

Also, I was wondering whether I was right in saying that (A ∪ B) ∩ (A ∪ ~B) is equivalent to (A - B) ?

**Annatala** · December 2nd 2011, 06:24 PM

The only thing I've ever seen big-omega used for in set theory is as another name for omega-1 (first uncountable ordinal). Does it also mean "universal set" in the context of intro set theory, then?

**Nombredor** · December 2nd 2011, 06:27 PM

Originally Posted by Annatala

Does it also mean "universal set" in the context of intro set theory, then?

Yes, that's what I meant it as. I've also seen people use just ''u'', but I prefer omega because ''u'' also looks a lot like the reunion symbol.

**Nombredor** · December 2nd 2011, 06:27 PM

double post

**Amer** · December 2nd 2011, 06:58 PM

Originally Posted by Nombredor

Also, I was wondering whether I was right in saying that (A ∪ B) ∩ (A ∪ ~B) is equivalent to (A - B) ?

it is correct

**Plato** · December 3rd 2011, 05:55 AM

Originally Posted by Nombredor

Also, I was wondering whether I was right in saying that (A ∪ B) ∩ (A ∪ ~B) is equivalent to (A - B) ?

No that is not correct.
Let $\Omega =\{1,2,3,4,5,6,7,8,9\},~A=\{1,2,3,4\}~\&~B=\{2,4,6 ,8\}$ .

Find $(A\cup B)~\&~(A\cup B^c)$ .

What is their intersection? What is $A-B~?$

**Amer** · December 3rd 2011, 07:28 AM

it is equal to A

**Plato** · December 3rd 2011, 07:37 AM

Originally Posted by Amer

it is equal to A

Correct. So the OP

Originally Posted by Nombredor

I was right in saying that
(A ∪ B) ∩ (A ∪ ~B) is equivalent to (A - B) ?

Is incorrect.

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