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I need to prove that $f(A_0 \cup A_1) = f(A_0) \cup f(A_1)$ where $f: A \to B$ is a mapping of sets and $A_0,A_1 \subseteq A$.

So, I saw this question and figured out the first direction ($f(A_0 \cup A_1) \subseteq f(A_0) \cup f(A_1)$). I got a little stuck in the other direction and I wanted to know if I worded my proof logically:

Let $y \in f(A_0 \cup A_1)$ be arbitrary. Then $y \in f(A_0 \cup A_1) = \{f(a) | a \in A_0 \cup A_1\}$ means that $y = f(x)$ for some $x \in A_0 \cup A_1$. By the definition of set union $x \in A_0 \cup A_1$ means $x \in A_0$ or $x \in A_1$. Recall $y = f(x)$ and suppose $x\in A_0$, it follows that $y \in \{f(a) | a \in A_0\} = f(A_0)$. Now, suppose instead that $x \in A_1$ and $y = f(x)$, it follows that $y \in \{f(a) | a\in A_1\} = f(A_1)$. Thus, in either case, $y \in f(A_0) \cup f(A_1)$. Since $y \in f(A_0 \cup A_1)$ and we have just shown that $y \in f(A_0) \cup f(A_1)$ then $f(A_0 \cup A_1) \subseteq f(A_0) \cup f(A_1)$.

Let $y \in f(A_0) \cup f(A_1)$ be arbitrary. Then $y \in f(A_0) = \{f(a)|a \in A_0\}$ or $y \in f(A_1) = \{f(a)|a \in A_1\}$. So $y \in f(A_0)$ means $y = f(x)$ for some $x \in A_0$ or $y = f(x)$ for some $x \in A_1$. By the definition of set union $x \in A_0 \cup A_1$, thus $y \in f(A_0 \cup A_1)$. Therefore, $f(A_0 \cup A_1) = f(A_0) \cup f(A_1)$.

I think I'm missing the connection between defining the set union and getting to the desired result.

Rick
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TheMathochist
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1 Answers1

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This looks fine to me except for one point in your proof for the other inclusion, namely:

So $y \in f(A_0)$ means $y = f(x)$ for some $x \in A_0$ or $y = f(x)$ for some $x \in A_1$.

$y \in f(A_0)$ means $y = f(x)$ for some $x \in A_0$, but not that $y = f(x)$ for some $x \in A_1$ (unless $A_0 \subseteq A_1$). What you want to say is something like:

$y \in f(A_0)$ means $y = f(x)$ for some $x \in A_0$, and $y \in f(A_1)$ means $y = f(x)$ for some $x \in A_1$. In either case, since $y \in f(A_0) \cup f(A_1)$ we have that $y = f(x)$ for some $x \in A_0$ or for some $x \in A_1$, so by definition of set union $x \in A_0 \cup A_1$.

Rick
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