Are the cardinalities equal?

Asked Jun 10 '20 at 01:14

Active Jun 10 '20 at 03:54

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Let $A$ be an uncountable set. Let $C$ be a countable set with $A\cap C = \emptyset$. Show that $|A\cup C| = |A|$.

Since $A$ is uncountable, it has an infinite countable subset, say $B$. Notice that $A \cup C = (A-B) \cup (B \cup C)$ and $(A-B)\cap (B \cup C) = \emptyset$. Since $B$ is an infinite countable set, $|B|=|\mathbb{N}|$. Since the union of countable sets is countable, and since $B$ is an infinite countable set, $B \cup C$ is an infinite countable set, so $|B \cup C|=|B| = |\mathbb{N}|$. Therefore, there exists a bijection $f:B\cup C \to B$. Define $h:A \cup C \to A$ by \begin{align*} h(x) = \begin{cases} f(x), & x \in B \cup C\\ x, & x \in A - B. \end{cases} \end{align*} From here I just need to show that $h$ is injective and surjective. Is this the right approach? Or am I missing any details?

edited Jun 10 '20 at 03:54

asked Jun 10 '20 at 01:14

INQUISITOR

6

This is a correct approach. – Ethan Bolker Jun 10 '20 at 01:17
1

I don't think the general-topology tag is appropriate here -- maybe elementary-set-theory instead? – MacRance Jun 10 '20 at 01:21
Does this answer your question? Prove that $|AUC| = |A|$, where $A$ is an uncountable set and $C$ is a countable set. – Sahiba Arora Jun 10 '20 at 09:05

Are the cardinalities equal?

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