Unions and Intersections of Sets (Ambiguous set?)

First off, I hope this question is in the right place. I know this isn't a big part of Calculus and is probably review of something I've not had, but I don't know where else to place it because this is currently what I'm learning in Calculus. I'm currently taking an online Calculus class and I am absolutely trying my hardest, but I'm having a lot of issues. I have the following homework problem and I have several parts of it answered but I need help with the parts that will be in bold/blue.

1.2.1 (L)
(a) Show that (A n B) u C and A n (B u C) need not be equal sets.

The easiest way to do this is by plugging in numbers, so:
Let A= {1, 2}, B= {2, 3}, C= {1, 3}
(A n B) u C ≠ A n (B u C)
(A n B)= {2}, C= {1, 3}
àu= {1, 2, 3}
A n (B u C)
B u C= {1, 2, 3}
A= {1, 2}
àn= {1, 2}
(A n B) u C ≠ A n (B u C) à {1, 2, 3}≠{1,2}
Since the same numbers are contained within both sets, the sets are still relatable. But the sets do not contain the same amount of numbers, so they are not equal.

(b)What is the most general case in which (A n B) u C and A n ( B u C)will be equal?
The most general case where (A n B) u C and A n (B u C) are equal is when:
A=B=C= {}
But another case could be A=B=C= {1, 2, 3}

(c)Why is it ambiguous to write A n B u C?
It is ambiguous to write this because it does not show any independent events

(d)Is it ambiguous to write A n B n C?

I understand that (C) is ambiguous because it doesn't show any independent events and I can visualize it and understand it in my head, but I'm having a hard time putting it on paper. Furthermore, I'm not sure if he wants us to show how it's ambiguous or if he just wants us to explain it. If there's anyway I can show it, let me know.

With part (D) I'm going to assume that it's not ambiguous because all are intersections of and will all yeild the same results, but again if I have to show my work of how I came to that conclusion, I can't because I don't know.

Any help is appreciated. Someone helped me with this problem earlier on this forum and I really appreciate it. I think I'm starting to catch on but it's hard. It's almost like trying to teach myself a foreign language.

samistumbo said:

(d)Is it ambiguous to write A n B n C ?

Click to expand...

I am thinking if you can show that


\(\displaystyle (A \cap B) \cap C \ = \ (A \cap C) \cap B \ = \ (B \cap C) \cap A, **\)

\(\displaystyle then \ \ A \cap B \cap C \ \ is \ not \ ambiguous.\)


If any (at least one) of the three sets is the empty set,
then those three cases simplify to the empty set, apiece.
Do you know how/why?


Else, let's suppose each set is non-empty. Suppose there
is a group of elements for each set that are members only
to that set and no others. And suppose that when the sets
are taken in pairs, they have a distinct group of elements
common just to those pairs, respectively. And suppose that
there is one group of elements that is common to all three
sets.


Let a, b, c, w, x, y, k represent groups of elements that
satisfy the conditions given in the previous paragraph.



Let A = {a, w, x, k}

Let B = {b, w, y, k}

Let C = {c, x, y, k}



\(\displaystyle Simplify \ each \ of \ the \ three \ cases \ in \ ** \)

\(\displaystyle to \ see \ if \ you \ arrive \ at \ the \ same \ resultant \ set \)

\(\displaystyle for \ all \ three \ cases.\)