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I am reading Enderton's book Elements of set theory and I am having hard time understanding Infinite cartesian product section in which he define's product of $H(i)$'s.

Could you please explain in other simple words or by giving an example.

PS : I am a self learner

Thanks. enter image description here

Mauro ALLEGRANZA
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Jasser
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  • Probably a bunch more have explanations. Search the site. – Asaf Karagila May 22 '18 at 06:47
  • Start from the cartesian product. What is $A \times B$ ? The set of pairs where the first "coordinate" belongs to $A$ and the second one belongs to $B$ : $A \times B = { (a,b) \mid a \in A \text { and } b \in B }$. We can iterate it for $n$ finite whatever : every element of $A_1 \times \ldots \times A_n$ will be an $n$-uple for which the "$i$-th coordinate" belongs to $A_i$. – Mauro ALLEGRANZA May 22 '18 at 06:49
  • Now, instead of the $A_i$'s, consider a set $I$ whatever and a "family" of sets ${ H(i) }$ : we say that the family is "indexed by $I$". We generalize the above construction in this way : the members of $\Pi_{i \in I} H(i)$ are "$I$-tuples" for which the "$i$-th coordinate" (i.e., the value at $i$) is an element of $H(i)$. – Mauro ALLEGRANZA May 22 '18 at 06:51
  • thanks @MauroALLEGRANZA but what about the part where he says {f | f is a function with domain I and for every i in I f(i) belongs to H(i)} – Jasser May 22 '18 at 07:02
  • In set theory an $n$-uple $(a_1, \ldots, a_n)$ of elements of $A$ is a function $f : { 1,2,\ldots, n } \to A$. Thus the $I$-uple is a function $f$ with domain $I$ sich that every "value" $f(i)$ belongs to $H(i)$. – Mauro ALLEGRANZA May 22 '18 at 07:07
  • "infinite elements" ???? The trick of the construction is exactly to avoid "listing" the elements of $I$ (compare with the finite case : ${ 1,2,\ldots, n }$). Thus, it works also for $I$ infinite. – Mauro ALLEGRANZA May 22 '18 at 07:24
  • I get it now. Once we index H with I than there should not be any problem even if I is infinite. Thanks for the help. really appreciate it @MauroALLEGRANZA – Jasser May 22 '18 at 10:45

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