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If $S_1$ and $S_2$ are both subsets of some arbitrary set $A$, then how do I prove that $f(S_1\cup S_2) = f(S_1) \cup f(S_2)$ for ALL cases

I understand that it is true, but I don't know how to prove it.

Stefan Hamcke
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user105750
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2 Answers2

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If $y\in f(S_1\cup S_2)$ then there is $x\in S_1\cup S_2$ such as $y=f(x)$.

If $x\in S_1\cup S_2$ then $x\in S_1$ or $x\in S_2$.

If $x\in S_1$ then $y=f(x)\in f(S_1)$.

Now you can finish.

Carlos Eugenio Thompson Pinzón
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if I understand your question correctly, you're given some function $f:S\rightarrow T$ for two sets $S, T$. You also have two subsets $S_1, S_2$ of S. Then you're asking if the image of the union of the two subsets is the union of the images? If this is your question, then suppose that $y$ is in the image of the unions, then it's in the image of one of the two subsets, so there is an $x$ in, say $S_1$ that f sends $x$ to. This $x$ is in the union of the two subsets. Now, if $y$ is just in the image of the union, there is some $x$ in the union that f sends to y. This $x$ is nessacarilly in one of the two subsets, by definition of union.

RougeSegwayUser
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