Math Help - Question about sets

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

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 $\Omega$.

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 ?

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 $B^c~.$

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

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.

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 ?

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

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?

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.

double post

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

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 $A-B~?$

it is equal to A