# Halmos naive set theory families

Halmos writes...

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 an indexed set, the function itself is called a family, and the value of the function x at an index i, called the term of a family, is denoted by x small i.

Say I = {1,2,3} and X = {8,9,10} and x small i = i + 3. So the indexed set would be {4,5,6}.

An unacceptable but generally accepted way of communicating the notation and indicating the emphasis is to speak of a family {x small i} in X, or of a family {x small i) of whatever the elements of X may be.

Is he saying the indexed set is a subset of X?

Further down the page he writes...

if I = 2, so that the range of the family {A small i} is the unordrerd pair {A small 0, A small 1}, then U small i A small i = a small0 U A small 1.

Why is I = 2 and not a set?

Thanks in advance

• "Why is I = 2 and not a set?" But $2= \{ 0,1 \}$ is a set. See page 44. – Mauro ALLEGRANZA Sep 4 '18 at 10:42
• It's supposed to be the case that $x$ is a function $I\to X$, so your example is weird because with $I$ and $X$ as you've defined them, $x_i=i+3$ is not a function $I\to X$. So yes, the set indexed by $x$ is supposed to be a subset of $X$. – Malice Vidrine Sep 4 '18 at 10:50
• "Say $I = \{ 1,2,3 \}$ and $X = \{ 8,9,10 \}$ and $x_i = i + 3$". NO; $x_i$ must be an element of $X$. Thus we may have $x= \{ (1,8),(2,9),(3,10) \}$; in this way: $x_1=8, x_2=9, x_3=10$. – Mauro ALLEGRANZA Sep 4 '18 at 11:12
• Okay. So why in my other question (math.stackexchange.com/questions/2902385/…) did people say x needed to be assigned some value? – Victor Sep 4 '18 at 11:36
• $x$ is a function, i.e. a set. You can define it either explicitly, as in my example : $x = \{ (1,8),(2,9),(3,10) \}$, where $I= \{ 1,2,3 \}$ and $X= \{ 8,9,10 \}$, or through a formula (as in the answer linked) : $x : I \to \mathbb N$ with again $I= \{ 1,2,3 \}$ and $x(i)=i+3$, in which case $x= \{ (1,4), (2,5), (3,6) \}$ and the range of $x$ (the indexed set) will be $X= \{ 4,5,6 \}$. The two are not the same function. – Mauro ALLEGRANZA Sep 4 '18 at 11:47

## 1 Answer

"Is he saying the indexed set is a subset of X?"

Yes.

We have that $x_i \in X$. Thus, according to the "unacceptable" view, the "family" $\{ x_i \} \subseteq X$.

The formal definition is :

that $x$ is a function from a set $I$ to a set $X$ [i.e. $x : I \to X$]. 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 an index $i$, called a term of the family, is denoted by $x_i$.

$x$ being a function, it will be a set of pairs :

$x= \{ (i,a) \mid i \in I \text { and } a \in X \}$.

Thus we have $(i,a) \in x$.

But being $x$ a function, we can consider also the "functional" notation [see page 31] : $x(i)$, abbreviated : $x_i$.

Obviously, the value of the function $x$ for argument $i$ (the term) will be an element of $X$, and thus we have : $x_i \in X$.

So the range of the function (called an indexed set) is :

$\{ x_i \} = \{ a \in X \mid (i,a) \in x, \text { for some } i \in I \} \subseteq X$.