About index (set theory)

Question

Definition: A collection of sets $\mathbb{E} $ is said to be indexed by a set A if and only if there is a funcitom F from A onto $\mathbb{E} $. In this case we call A the index set lf $\mathbb{E} $, say $\mathbb{E} $ is indexed by A, and represent F(a) by $ E_a $. In particular, $\mathbb{E} $ is indexed by A means $\mathbb{E} =\{E_a\}_{a \in A} $.

I am not sure how to draw up the relation between function and sets in this case. Can someone explain this definition about index more intuitively? Thanks.

Answer 1

Suppose that we have sets $\{a,b,c\}$, $\{a,d,x,y\}$, and $\{c,u\}$; we can form the collection

$$\Bbb E=\big\{\{a,b,c\},\{a,d,x,y\},\{c,u\}\big\}\;.$$

Now let $A=\{1,2,3\}$; we can define a function $f:A\to\Bbb E$ by

$$\begin{align*} f(1)&=\{a,b,c\}\\ f(2)&=\{a,d,x,y\}\\ f(3)&=\{c,u\}\;. \end{align*}$$

The collection $\Bbb E$ is now indexed by $A$ via the indexing function $f$, and instead of having to talk about $\{a,b,c\}$, $\{a,d,x,y\}$, and $\{c,u\}$, we can refer to the three sets by their indices: $E_1=f(1)=\{a,b,c\}$, $E_2=f(2)=\{a,d,x,y\}$, and $E_3=f(3)=\{c,u\}$.

Of course we could index $\Bbb E$ by other sets. We might, for instance, use the set $I=\{0,1,2\}$, with the indexing function $g$ defined by $g(0)=\{c,u\}$, $g(1)=\{a,b,c\}$, and $g(2)=\{a,d,x,y\}$; in that case we would refer to the sets as $E_0=g(0)=\{c,u\}$, $E_1=g(1)=\{a,b,c\}$, and $E_2=g(2)=\{a,d,x,y\}$.

An index set $A$ for $\Bbb E$ is just a set of convenient labels for the members of $\Bbb E$, and the indexing function $f$ simply assigns each label in $A$ to one of the members of $\Bbb E$ in such a way that each member of $\Bbb E$ gets a label. If $a\in A$ is assigned to some $X\in\Bbb E$, i.e., if $f(a)=X$, we’ve given $X$ the label $a$ and can now call it $E_a$. This brings some uniformity into the names that we’re using for the members of $\Bbb E$.

Answer 2

This definition is essentially saying that we can identify an element of $\mathbb{E}$ with its index, where its index is an element of $A.$ The notation $E = \{E_{\alpha}\}_{\alpha \in A}$ shows this nicely. Suppose that $E_{\alpha} \in \mathbb{E}.$ The definition of index is showing that there exists some element of $A,$ $\alpha,$ such that (using your function $F : A \to \mathbb{E}$) $F(\alpha) = E_{\alpha}.$ Thus, the index is taking a set and identifying (using the function $F$) an element of the index set with this element of $\mathbb{E}.$ The reason this function must be surjective is so we can "index" each element (i.e. there is some element in $A$ that we can identify to a given element of $\mathbb{E}$).