 # Problem Set #1

### Objective

During this activity, students should be able to:

• Write some relatively simple functions using the Clojure programming language.

This activity helps students develop the following skills, values and attitudes: ability to analyze and synthesize, capacity for identifying and solving problems, and efficient use of computer systems.

## Activity Description

This activity can be developed individually or in pairs.

Solve the following programming problem set using Clojure. Place all your functions and unit tests in a file called problemset1.clj.

1. The function ! takes a positive integer n as its argument and returns its corresponding factorial, which is mathematically defined as:

$$n!=\begin{cases} 1 & \text{ if } n= 0\\ n \cdot (n-1)! & \text{ if } n>0 \end{cases}$$

Unit tests:

(deftest test-!
(is (= 1
(! 0)))
(is (= 120
(! 5)))
(is (= '(1 1 2 6 24 120 720 5040 40320 362880 3628800)
(map ! (range 11))))
(is (= 15511210043330985984000000N
(! 25)))
(is (= 815915283247897734345611269596115894272000000000N
(! 40))))

2. The function duplicate takes a list lst as its argument and returns a new list with each element of lst duplicated. Unit tests:
(deftest test-duplicate
(is (= '(1 1 2 2 3 3 4 4 5 5)
(duplicate '(1 2 3 4 5))))
(is (= ()
(duplicate ())))
(is (= '(a a)
(duplicate '(a))))
(is (= '(a a b b c c d d e e f f g g h h)
(duplicate '(a b c d e f g h)))))

3. The function pow takes two arguments as input: a number a and a positive integer b. It returns the result of computing a raised to the power b. Unit tests:
(deftest test-pow
(is (= 1 (pow 0 0)))
(is (= 0 (pow 0 1)))
(is (= 1 (pow 5 0)))
(is (= 5 (pow 5 1)))
(is (= 125 (pow 5 3)))
(is (= 25 (pow -5 2)))
(is (= -125 (pow -5 3)))
(is (= 1024 (pow 2 10)))
(is (= 525.21875 (pow 3.5 5)))
(is (= 129746337890625 (pow 15 12)))
(is (= 3909821048582988049 (pow 7 22))))

4. The function fib takes a positive integer n as its argument and returns the corresponding element of the Fibonacci sequence, which is mathematically defined as:

$$\text{fib}(n)=\begin{cases} n & \text{ if } n \le 1 \\ \text{fib}(n-1) + \text{fib}(n-2) & \text{ if } n>1 \end{cases}$$

Unit tests:

(deftest test-fib
(is (= 0
(fib 0)))
(is (= 1
(fib 1)))
(is (= 1
(fib 2)))
(is (= 5
(fib 5)))
(is (= '(0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610
987 1597 2584 4181 6765)
(map fib (range 21))))
(is (= 267914296
(fib 42))))

5. The function enlist surrounds in a list every upper-level element of the list it takes as input. Unit tests:
(deftest test-enlist
(is (= () (enlist ())))
(is (= '((a) (b) (c)) (enlist '(a b c))))
(is (= '(((1 2 3)) (4) ((5)) (7) (8))
(enlist '((1 2 3) 4 (5) 7 8)))))

6. The function positives takes a list of numbers lst as its argument, and returns a new list that only contains the positive numbers of lst. Unit tests:
(deftest test-positives
(is (= () (positives '())))
(is (= () (positives '(-4 -1 -10 -13 -5))))
(is (= '(3 6) (positives '(-4 3 -1 -10 -13 6 -5))))
(is (= '(4 3 1 10 13 6 5) (positives '(4 3 1 10 13 6 5)))))

7. The function add-list returns the sum of all the elements of its input list, or 0 if its empty. Assume that all the elements in the input list are numbers. Unit tests:
(deftest test-add-list
(is (= 10 (add-list '(2 4 1 3))))
(is (= 55 (add-list '(1 2 3 4 5 6 7 8 9 10)))))

8. The function invert-pairs takes as an argument a list of vectors containing two elements each. It returns a new list with every vector pair inverted. Unit tests:
(deftest test-invert-pairs
(is (= () (invert-pairs ())))
(is (= '([1 a][2 a][1 b][2 b]))
(invert-pairs '([a 1][a 2][b 1][b 2])))
(is (= '([1 January][2 February][3 March])
(invert-pairs '([January 1][February 2][March 3])))))

9. The function list-of-symbols? takes a list lst as its argument. It returns true if all the elements (possibly zero) contained in lst are symbols, or false otherwise. Use the symbol? predicate to determine if something is a symbol. Unit tests:
(deftest test-list-of-symbols?
(is (list-of-symbols? ()))
(is (list-of-symbols? '(a)))
(is (list-of-symbols? '(a b c d e)))
(is (not (list-of-symbols? '(a b c d 42 e))))
(is (not (list-of-symbols? '(42 a b c)))))

10. The function swapper takes three arguments as input: two values x and y, and a list lst. It returns a new list in which every occurrence of x in lst is swapped with y, and vice versa. Any other element of lst remains the same. You may assume that lst does not contain nested sequences. Unit tests:
(deftest test-swapper
(is (= ()
(swapper 1 2 ())))
(is (= '(4 3 4 9 9 3 3 3 9 9 7 9
3 7 8 7 8 4 5 6)
(swapper 1 2 [4 3 4 9 9 3 3 3 9 9 7
9 3 7 8 7 8 4 5 6])))
(is (= '(4 4 5 1 4 8 1 5 6 4 5 2 9 5 9 9 2 1 1 4)
(swapper 1 2 [4 4 5 2 4 8 2 5 6 4 5
1 9 5 9 9 1 2 2 4])))
(is (= '(soft purr warm purr little ball of fur
happy purr sleepy purr kitty kitty kitty)
(swapper 'purr
'kitty
'(soft kitty warm kitty little ball
of fur happy kitty sleepy kitty
purr purr purr)))))

11. The function dot-product takes two arguments: the lists a and b. It returns the result of performing the dot product of a times b. The dot product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products. Unit tests:
(deftest test-dot-product
(is (= 0 (dot-product () ())))
(is (= 32 (dot-product '(1 2 3) '(4 5 6))))
(is (= 21.45 (dot-product '(1.3 3.4 5.7 9.5 10.4)
'(-4.5 3.0 1.5 0.9 0.0)))))

12. The function average takes a list lst as its argument. It returns the arithmetic mean of the numbers contained in lst, or nil if lst is empty. The arithmetic mean ($$\bar{x}$$) is defined as:

$$\bar{x} = \frac{1}{n}\sum_{i=1}^{n}x_i$$

Unit test:

(deftest test-average
(is (nil? (average ())))
(is (= 4
(average '(4))))
(is (= 3
(average '(5 6 1 6 0 1 2))))
(is (= 2.5
(average '(1.7 4.5 0 2.0 3.4 5 2.5 2.2 1.2)))))

13. The function standard-deviation takes a list lst as its argument. It returns the population standard deviation of the numbers contained in lst, or nil if lst is empty. The population standard deviation ($$\sigma$$) is defined as:

$$\sigma = \sqrt{\frac{1}{n} \sum_{i=1}^{n}(x_i - \bar{x})^2}$$

Unit tests:

(deftest test-standard-deviation
(is (nil? (standard-deviation ())))
(is (aprox= 0.01
1.87
(standard-deviation
'(6 2 3 1))))
(is (aprox= 0.0001
12.3153
(standard-deviation
'(4 8 15 16 23 42))))
(is (aprox= 0.00001
7.07106
(standard-deviation
'(110 105 90 100 95))))
(is (aprox= 0.001
2.983
(standard-deviation
'(9 2 5 4 12 7 8 11
9 3 7 4 12 5 4 10
9 6 9 4)))))


NOTE: The previous test requires comparing floating point numbers. In order to avoid rounding errors, we need first to define the following function called aprox= (approximately equal):

(require '[clojure.math.numeric-tower :refer [abs]])

(defn aprox=
"Checks if x is approximately equal to y. Returns true
if |x - y| < epsilon, or false otherwise."
[epsilon x y]
(< (abs (- x y)) epsilon))


## Deliverables

The program source file must include at the top the authors’ personal information (name and student id) within comments. For example:

;----------------------------------------------------------
; Problem Set #1
; Date: September 13, 2019.
; Authors:
;          A01166611 Pepper Pots
;          A01160611 Anthony Stark
;----------------------------------------------------------


Also, each function should include a documentation string (docstring) with a brief description of its behavior. For example:

(defn max2
"Returns the largest of the two numbers x and y."
[x y]
(if (> x y) x y))


To deliver the problemset1.clj file, please provide the following information: