# What does this mean Y = {u : (∃z ∈ X) u ∈ z} = U {z: z ∈ X} = UX? I don't understand the property. [closed]

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 $$u ∈ z$$.

• Perhaps you should try some examples. For example, what is $\cup \{\{1, 2\}, \{3, 5, 0\}, \{0\}\}$? Jul 20 at 3:54
• What "property"? I don't see anything, just the definition of union. Jul 20 at 3:56
• @azif00 the property is (∃z ∈ X) u ∈ z}. Jul 20 at 3:58
• @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? Jul 20 at 4:07
• @MarkSaving I understand that, but what does u∈z mean in this context. Jul 20 at 4:09

$$Y$$ 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 $$X$$.