Induction And Images Of Unions

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Problem 82: (Induction And Images Of Unions)

Consider the function $f:X\to Y$. We already know from the image and preimage properties theorem that if $A_1$ and $A_2$ are subsets of $X$, then we have $f(A_1\cup A_2) = f(A_1)\cup f(A_2)$. Use induction to prove that for every $n\in \mathbb{N}$ if $A_1, A_2, \ldots, A_n$ are subsets of $X$, then we have $$f(A_1\cup A_2\cup \cdots \cup A_n) = f(A_1)\cup f(A_2)\cup \cdots\cup f(A_n).$$ A similar argument will prove that most of the other properties in the theorem can be extend from two sets to $n$ sets.

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