I stuck on something. I am trying to understand what is said in the image I linked above.
"A function I from a set Λ onto a set a is said to index the set a by Λ. The set Λ is called the index and a is the indexed set. If I(λ) = a, then we write aλ for I(λ)."
There's no examples and further explanations in the textbook. As far as I understood, Λ can be a set {1,2,3,4…} or, say {a,b,c,d..} and the set a can be {tree, fox, grass}. So after set a got indexed by the set Λ = {1,2,3}, we get a = {tree(1), fox(2), grass(3)} (Sorry, I don't know how to type so that "1" appear under the word "tree")
Am I right? And by the way, what is it supposed to mean: "If I(λ) = a, then we write aλ for I(λ)." I swear there's no λ mentioned before that line in the textbook. Is λ a variable that might represent one of those elements of {1,2,3…}? Say I could say that λ = 1 or λ = 15, right?
Explain please what is this all about. Thank you in advance.
Best Answer
In many situations it's handy to have a way for referring to elements of a set in a “uniform way”.
Think to this “indexing” as a way to name the elements of the set: it's quite similar to identifying a car with its plate number.
Suppose the set is $a=\{\mathrm{tree},\mathrm{fox},\mathrm{grass}\}$. The idea is to pick a surjective function from some set to $a$. The set could be $\{1,2,3\}$ or $\{\mathrm{Larry},\mathrm{Moe},\mathrm{Curly}\}$, but there's no restriction to the size of $I$, so long as the function we pick is surjective.
Remember that the set $a$ can also be described as $a=\{\mathrm{grass},\mathrm{fox},\mathrm{tree}\}$, so even if we choose $I=\{1,2,3\}$, there's no “canonical” indexing.
Both functions $$ \begin{cases} 1\mapsto\mathrm{tree}\\ 2\mapsto\mathrm{fox}\\ 3\mapsto\mathrm{grass} \end{cases} \qquad\text{and}\qquad \begin{cases} 1\mapsto\mathrm{grass}\\ 2\mapsto\mathrm{fox}\\ 3\mapsto\mathrm{tree} \end{cases} $$ are valid as an indexing of $a$. Also the function $\{1,2,3,4\}\to a$ defined by $$ \begin{cases} 1\mapsto\mathrm{tree}\\ 2\mapsto\mathrm{fox}\\ 3\mapsto\mathrm{grass}\\ 4\mapsto\mathrm{fox} \end{cases} $$ would be a valid indexing.
The indexing function is usually “unnamed”, so it's denoted by $\lambda\mapsto a_\lambda$, because its exact definition is mostly irrelevant. The important thing is that, for every $x\in a$, there exists $\lambda\in I$ such that $x=a_\lambda$, so that $x$ has a “plate number” (maybe more than one).