MHB Define Sets $\{x,y\}$ and $x \cup y$

[evinda]

Hey! (Wave)

If $x \neq y $, define the sets $\bigcup \langle x,y \rangle , \bigcup \bigcup \langle x,y \rangle$.

According to my notes, it is like that:

$$\langle x,y \rangle= \{ \{x\}, \{x,y\} \} $$

$$ \bigcup \langle x,y \rangle=\{x,y\} $$

$$ \bigcup \bigcup \langle x,y \rangle=x \cup y$$

Why is it $ \bigcup \bigcup \langle x,y \rangle=x \cup y$ and not $\bigcup \bigcup \langle x,y \rangle=\{x,y\}$ ? (Thinking)

 

[Evgeny.Makarov]

It follows from the definition of generalized union that $\bigcup\{A_1,\dots,A_n\}=A_1\cup\dots\cup A_n$. I would claim that this equality is the idea behind the definition of the generalized union, and this definition is a generalization to the case when the set is not necessarily finite.

 

[evinda]

It follows from the definition of generalized union that $\bigcup\{A_1,\dots,A_n\}=A_1\cup\dots\cup A_n$. I would claim that this equality is the idea behind the definition of the generalized union, and this definition is a generalization to the case when the set is not necessarily finite.

I understand..Thank you very much! (Smile)