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

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

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

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

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

You simply need to show that $\displaystyle P_1,P_2$ 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 $\displaystyle B_1$ and $\displaystyle B_2$ are (1) nonempty, (2) non-overlapping subsets and (3) $\displaystyle B_1\cup B_2 = B$, you know they are partitions of of the set $\displaystyle B$. These 3 conditions are what partition means; for example gender partitions humanity.