Partitions homework question (Probability course)

Question

The question states the following

Assume the sets $A$ and $B_{1},B_{2},...B_{n}$ are subsets of the universe. Assume also that $B_{1},B_{2},...B_{n}$ form a partition of the universe. Now define the following $E_{i} = A \cap B_{i}$ for $i = 1,2,.....n$ and $E_{n+1} = A^c$ Show that $E_{1}, E_{2}...., E_{n+1}$ also form a partition of the universe.

This is all I have reasoned so far:

1. We know that $B_{1},B_{2},...B_{n}$ are mutually exclusive events as they form a partition of the universe, it is given that they are subsets of the universe as well.
2. $A$ is also a subset of the universe, thus it must be somewhere within the unions of $B_{1},B_{2},...B_{n}$.
3. $E_{i}$ is the intersection of A with the $B_{i}$'s for $i$ from 1 to $n$.
4. Since $E_{n+1}$ is the complement of A this accounts for all of the missing parts in the universe that $E_{i} = A \cap B_{i}$ does not cover. Thus we can say that $E_{1}$ to $E_{n+1}$ covers the entire universe.
5. Since our sets of $E$ cover the entire universe, the intersection of $A$ with $B_{i}$ are mutually exclusive, along with the complement of A being mutually exclusive to all the $B's$ intersected with $A$, we can say that the $E's$ form a partition of the universe.

Is this enough to show that it forms a partition, this was my though process. How do I show this mathematically, I can explain the proof but I have trouble representing what I said above mathematically if it is even correct logic in the first place.

Any explanations or tips would be greatly appreciated.

Thanks

Answer 1

The "simple" visualization, which you mostly have, is that the first $E_1$ through $E_n$ form a partition of $A$, so they are pairwise disjoint and their union is $A$. Throw in $A^C$ and you get another pairwise disjoint set, and $A\cup A^C$ is the universe.

Answer 2

For $1\leq i < j \leq n$, $E_i \cap E_j = (A \cap B_i) \cap (A \cap B_j) = A\cap (B_i \cap B_j) = A \cap \emptyset = \emptyset $.

Also, $E_{n+1} \cap E_i = A^c \cap (A\cap B_i) = (A^c \cap A) \cap B_i = \emptyset \cap B_i = \emptyset $. Thus the sets $E_1, E_2, \dots , E_n, E_{n+1}$ are mutually exclusive.

Now, we have to show $ \bigcup_{i=1}^{n+1} E_i $ is the universal set $U$.

$ \displaystyle{\bigcup_{i=1}^{n+1} E_i = A^c \cup \left(\bigcup_{i=1}^{n} (A \cap B_i) \right) = A^c \cup \left(A \cap \bigcup_{i=1}^{n} B_i \right) = A^c \cup (A\cap U)= A^c \cup A = U}$

and we are done.