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What does this mean $Y = \{u : \exists z \in X (u \in z)\} = \bigcup\{z: z \in X\} = \bigcup X$? I don't understand the property. The issue is with $u ∈ z$.

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  • $\begingroup$ Perhaps you should try some examples. For example, what is $\cup \{\{1, 2\}, \{3, 5, 0\}, \{0\}\}$? $\endgroup$ Jul 20 at 3:54
  • $\begingroup$ What "property"? I don't see anything, just the definition of union. $\endgroup$
    – azif00
    Jul 20 at 3:56
  • $\begingroup$ @azif00 the property is (∃z ∈ X) u ∈ z}. $\endgroup$ Jul 20 at 3:58
  • $\begingroup$ @JackFrosher That means "there exists some $z$ which is an element of $X$, such that $u \in z$". What deeper meaning are you trying to extract from it? $\endgroup$ Jul 20 at 4:07
  • $\begingroup$ @MarkSaving I understand that, but what does u∈z mean in this context. $\endgroup$ Jul 20 at 4:09
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$Y$ is the set of elements that are elements in some element in set $X$. $$Y=\{ u : \exists z\in X~.u\in z\}$$

More simply: $Y$ is the union of sets within $X$. $$Y=\bigcup X = \bigcup \{z: z\in X\} $$

Should $X$ be $\{\{1\},\{2,4\},\{1,3,4\}\}$, then because there do exists sets within $X$ that contain the elements in $\{1,2,3,4\}$ and no other, that set will be $Y$. This is clearly the union of sets within this $X$.


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  • $\begingroup$ I see. Thank you for your time. $\endgroup$ Jul 20 at 4:50

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