1. ## Inverse of sets

I have attached a question that I am having trouble with. Thanks to anyone who can help.DOC1.DOC

2. ## Sets and Inverse functions

Hello Mel

Here is the problem.

Suppose $\displaystyle X$ and $\displaystyle Y$ are nonempty sets and $\displaystyle f: X \rightarrow Y$ is a function. Let $\displaystyle A$ and $\displaystyle B$ be subsets of $\displaystyle Y$. Prove that $\displaystyle f^{-1} (A \cup B) = f^{-1}(A) \cup f^{-1}(B)$

We do this by showing that $\displaystyle f^{-1} (A \cup B) \subseteq f^{-1}(A) \cup f^{-1}(B)$ and $\displaystyle f^{-1}(A) \cup f^{-1}(B) \subseteq f^{-1} (A \cup B)$

So, for the first part:

Suppose $\displaystyle x \in f^{-1}(A \cup B)$. Then for some $\displaystyle y \in Y$, $\displaystyle f(x) = y$, and $\displaystyle y \in A \cup B$.

$\displaystyle \Rightarrow y \in A$ or $\displaystyle y \in B$ [Note: throughout this proof 'or' means 'inclusive or'. So $\displaystyle y \in A$ or $\displaystyle y \in B$ means $\displaystyle y \in A$ or $\displaystyle y \in B$ or both.

$\displaystyle \Rightarrow f(x) \in A$ or $\displaystyle f(x) \in B$

$\displaystyle \Rightarrow x \in f^{-1}(A)$ or $\displaystyle x \in f^{-1}(B)$

$\displaystyle \Rightarrow x \in f^{-1}(A) \cup f^{-1}(B)$

So $\displaystyle x \in f^{-1}(A \cup B) \Rightarrow x \in f^{-1}(A) \cup f^{-1}(B)$

$\displaystyle \Rightarrow f^{-1}(A \cup B) \subseteq f^{-1}(A) \cup f^{-1}(B)$ (1)

For the second part:

Suppose $\displaystyle x \in f^{-1}(A) \cup f^{-1}(B)$

Then $\displaystyle x \in f^{-1}(A)$ or $\displaystyle x \in f^{-1}(B)$

Then for some $\displaystyle y \in Y$, $\displaystyle f(x) = y$ and $\displaystyle y \in A$ or $\displaystyle y \in B$

$\displaystyle \Rightarrow y \in A \cup B$

$\displaystyle \Rightarrow x \in f^{-1}(A \cup B)$

So $\displaystyle x \in f^{-1}(A) \cup f^{-1}(B) \Rightarrow x \in f^{-1}(A \cup B)$

$\displaystyle \Rightarrow f^{-1}(A) \cup f^{-1}(B) \subseteq f^{-1} (A \cup B)$ (2)

So, from (1) and (2): $\displaystyle f^{-1} (A \cup B) = f^{-1}(A) \cup f^{-1}(B)$