Checking injectivity of a certain function from a union of a family indexed by $K$ to $K$

Question

Let $A = \{ A_k | k \in K \}$ be a family of sets indexed by $K$. By Zermelo's theorem, $K$ can be well-ordered. Now, let $\leq$ be a well-order on $K$.

Define

$j: \bigcup\limits_{k \in K} A_k \to K$,

$j(x)=\min_{\le}\{k\in K:x\in A_k\}$

Is this function injective? How can this be checked and proved or disproved?

Answer 1

Let $A_1 = \{1,2\}, A_2 = \{2,3\}$. Then $K = \{1,2,3\}$.

Then, $j(1) = 1$, $j(2) = 1$, so this is not injective.

Question for you: What if each $A_i$ contains at most 1 element? Then what can you say?

Answer 2

Take $K= \mathbb{N}$, $A_1= \left\{1,2 \right\}$, $A_2= \left\{3,4 \right\},...$. Then $j: \mathbb{N} \rightarrow \mathbb{N}$ and $j(1)=j(2)$, so no $j$ is not necessarily injective.