Let $A_1, \dots A_n$ be finite sets. How can we show that the $\bigcup_{k=1}^{n} A_k$ ,union of finite dimensional sets, is finite?
Finite union of finite sets is finite
elementary-set-theory
elementary-set-theory
Let $A_1, \dots A_n$ be finite sets. How can we show that the $\bigcup_{k=1}^{n} A_k$ ,union of finite dimensional sets, is finite?
Best Answer
We know that $$\left|\bigcup_{i=1}^n A_i \right| \le \sum_{i=1}^n |A_i|$$ If you want to show this, try a proof using induction. Since $|A_i| < \infty$ for every $i$, $\sum_{i=1}^n |A_i| < \infty$. So, $\left|\bigcup_{i=1}^n A_i \right| < \infty$. The proof is complete.
This result is pretty intuitive, and here are some thoughts on a proof by contradiction. Suppose $\left|\bigcup_{i=1}^n A_i \right| = \infty$. If you delete finitely many elements from an infinite set, you still have infinite elements (you might ask for a proof of this too, but you need to start somewhere). Hence, by removing elements from the union which correspond to elements from the sets $A_i$ for $1\le i\le n$, you should still be left with infinite elements. However, the definition of union tells me that $x \in \bigcup_{i=1}^n A_i$ if and only if $x \in A_i$ for some $i$ such that $1\le i\le n$. Contradiction!