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I am learning set theory and am currently trying to understand the chapter on families in Naive set theory by Halmos.

He writes:

Suppose, for instance that $x$ is a function from a set $I$ to a set $X$. An element of the domain $I$ is called an index, $I$ is called the index set, the range of the function is called an indexed set, the function itself is called a family, and the value of the function $x$ at and index $i$, called a term of the family, is denoted by $x_i$.

.......speak of a family $\{x_i\}$ in X.

Say I = {1,2,3} and X = {4,5,6} Is the family = {x1,x2,x3}? OR are there 3 families {x1},{x2},{x3}?

What am I missing here

Thanks,

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  • $\begingroup$ I don't see any function here. $\endgroup$ – Lord Shark the Unknown Sep 2 '18 at 6:26
  • $\begingroup$ x is a function $\endgroup$ – Victor Sep 2 '18 at 6:27
  • $\begingroup$ I see no function $x$ in your examples. $\endgroup$ – Lord Shark the Unknown Sep 2 '18 at 6:28
  • $\begingroup$ Say I = {1,2,3} and X = {4,5,6} Is the family = {x1,x2,x3}? OR are there 3 families {x1},{x2},{x3}?......there's 6 lol $\endgroup$ – Victor Sep 2 '18 at 6:33
  • $\begingroup$ $\{x_1,x_2,x_3\}$ doesn't mean anything unless you specify the function $x$ first. $\endgroup$ – Malice Vidrine Sep 2 '18 at 6:35
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Let's say $I=\{1,2,3\}$ and $X=\mathbb{N}$. If $x$ is the function defined on $I$ such that $x_i=i+3$ then, in the terminology you quote, $x$ is the family, $\{4,5,6\}$ is the set indexed by $I$.

Another example would be $I=\mathbb{N}$ and $X=\mathcal{P}(\mathbb{N})$ and $x_i=\{m:\exists y(y\in\mathbb{N}\wedge m=y^i)\}$. That is, the term $x_i$ at each index $i$ is the set of all natural numbers that are the $i$-th power of something.

Simply specifying $I$ and $X$ doesn't uniquely determine an indexed family; such families are an extra piece of information, namely a function $x:I\to X$.

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  • $\begingroup$ what do you mean i+3? $\endgroup$ – Victor Sep 2 '18 at 6:37
  • $\begingroup$ "$x_i$" is just another way to write $x(i)$, the application of the function $x$ to an element $i\in I$. $\endgroup$ – Malice Vidrine Sep 2 '18 at 6:38
  • $\begingroup$ So $x(i)=i+3$, is exactly the same sort of thing as $f(x)=x+3$, just a different choice of letters. $\endgroup$ – Malice Vidrine Sep 2 '18 at 6:40
  • $\begingroup$ but why the number 3? $\endgroup$ – Victor Sep 2 '18 at 6:45
  • $\begingroup$ It's arbitrary. It's just an example. $\endgroup$ – Malice Vidrine Sep 2 '18 at 6:46

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