Consider these definitions. What does the function f[] do? a[{}, e-]:={} a[1List, e] := Prepend[

Consider these definitions. What does the function f[] do?...

Consider these definitions. What does the function $\mathrm{f}[]$ do? $a\left[\{\}, e_{-}\right]:=\{\}$ a[1_List, e_] := Prepend[ a[Rest[1], e], Prepend[First[1], e] ] $f[\{\}]:=\{\{\}\}$ $f[1$ List $]:=\operatorname{Join}[a[f[\operatorname{Rest}[1]], \operatorname{First}[1]], f[\operatorname{Rest}[1]]]$ 1. What is $f[\{x\}], f[\{x, y\}]$ and $f[\{x, y, z\}]$ ? 2. If $\mathrm{f}$ is called with a list of $n \geq 0$ elements (i.e., in the form $\mathrm{f}\left[\left\{e_1, e_2, \ldots, e_n\right\}\right]$ ), what length does the result have? 3. Which well-known function is implemented by $\mathrm{f}\left[\left\{e_1, e_2, \ldots, e_n\right\}\right]$ ? Assume that all $e_i$ are pairwise distinct.

Consider these definitions. What does the function $\mathrm{f}[]$ do?
$a\left[\{\}, e_{-}\right]:=\{\}$
a[1_List, e_] := Prepend[ a[Rest[1], e], Prepend[First[1], e] ]
$f[\{\}]:=\{\{\}\}$
$f[1$ List $]:=\operatorname{Join}[a[f[\operatorname{Rest}[1]], \operatorname{First}[1]], f[\operatorname{Rest}[1]]]$
1. What is $f[\{x\}], f[\{x, y\}]$ and $f[\{x, y, z\}]$ ?
2. If $\mathrm{f}$ is called with a list of $n \geq 0$ elements (i.e., in the form $\mathrm{f}\left[\left\{e_1, e_2, \ldots, e_n\right\}\right]$ ), what length does the result have?
3. Which well-known function is implemented by $\mathrm{f}\left[\left\{e_1, e_2, \ldots, e_n\right\}\right]$ ? Assume that all $e_i$ are pairwise distinct.

Key Concepts

List Manipulation in Functional Programming

List manipulation is a fundamental aspect of functional programming. Operations such as adding an element to the front of a list (often called prepending) and concatenating lists are used to build and combine results. These techniques are particularly useful in constructing complex data structures like the power set from simpler parts.

Power Set

The power set of a set is the collection of all its subsets, including the empty set and the set itself. For a set containing n unique elements, the power set contains 2^n subsets. Generating the power set is a common combinatorial problem, often solved using recursive techniques that systematically consider whether to include each element in a subset.

Base Case and Recursive Step

In any recursive function, the base case provides a simple, direct answer for the smallest instance of the problem. The recursive step then solves larger instances by relying on the solutions of smaller instances. This pattern is crucial in ensuring that the recursion terminates and that the function correctly builds up the complete solution.

Recursion

Recursion is a programming technique where a function calls itself to solve a problem by breaking it down into simpler subproblems. In recursive algorithms, a clear base case is defined to stop the recursion, and each recursive call works on a smaller part of the input, eventually combining these solutions to solve the overall problem.

Transcript

00:02 Okay, so for this question, we're going to do several different composite functions.

00:05 So for the first one, i'm going to do the, i'm going to find f of 2, f sub 2 of f sub 3 of x, which means i'm going to put f sub 3 into f sub 2.

00:21 So i will have s sub 2 of f sub 3 equals 1 minus x from my list up top.

00:27 And now i'm putting that into f -sub -2 where the x is, so that will get me 1 over 1 -1 -x.

00:37 For the second one, we will be doing f -sub -3 of f -sup 5 of x.

00:48 So this is actually the same thing as saying f -sop -3, f -5 of x.

00:54 They're equivalent.

00:55 Either way, i'm putting f -sop -5 into f -sop -5.

00:58 Of 3.

01:01 F sub 5 is 1 over 1 minus x from my list up top.

01:07 Now i'm putting that into f 3 where the x is.

01:11 So i will have 1 minus 1 minus 1 over x.

01:16 So i now need to find a common denominator so i'm going to multiply 1 minus x to the 1 on the top and bottom.

01:22 So i will have 1 minus x minus 1 in my numerator all over 1.