1. ## set partitions

I've been stuck on this question for a while now and was hoping someone might be able to give me some help cause i don't really understand it.

Let B be a set. Define what it means to say that
B1,B2 give a partition of B, where B1 and B2 are subsets of B. Let
f : A −→ B be a function. Suppose that C is a subset of B. Write down
the definition of f−1(C). Suppose that B1,B2 is a partition of B. Prove,
using your definitions, that f−1(B1), f−1(B2) is a partition of A

2. $B_1,B_2$ is a partition of $B$ when:

1. $B_1\cap B_2 = \emptyset$
2. $B_1,B_2\neq \emptyset$
3. $B_1\cup B_2 = B$

Denote $B = B_1\oplus B_2$ as the disjunct union of $B_1,B_2$.

If $f:A\to B$ is a function and $C\subset B$. You can write $C = (C\cap B_1)\oplus(C\cap B_2)$. Then $f^{-1}(C) = f^{-1}(C\cap B_1\oplus C\cap B_2) = f^{-1}(C\cap B_1)\oplus f^{-1}(C\cap B_2) = P_1\oplus P_2$.
Then we've shown that $P_1,P_2$ form a partition of $f^{-1}(C)\subset A$.

The same way is shown that $f^{-1}(B) = f^{-1}(B_1)\oplus f^{-1}(B_2) = P_1\oplus P_2 = A$.

You simply need to show that $P_1,P_2$ satisfy conditions 1,2,3

3. Let B be a set. Define what it means to say that
B1,B2 give a partition of B, where B1 and B2 are subsets of B.
If you have a collection of nonempty subsets of B which are pairwise disjoint (any two distinct ones are disjoint), and the union of all these subsets is the set itself, then you know those subsets are partition of that set.

If $B_1$ and $B_2$ are (1) nonempty, (2) non-overlapping subsets and (3) $B_1\cup B_2 = B$, you know they are partitions of of the set $B$. These 3 conditions are what partition means; for example gender partitions humanity.