Exercises for Week 3: Lists and List Comprehensions
Here are some exercises designed to help you practice programming with lists
and list comprehensions.
You may need the following useful standard functions:
or :: [Bool] -> Bool
- returns
True
if any element of its argument list is True
.
and :: [Bool] -> Bool
- returns
True
if every element of its argument list is True
.
nub :: Eq a => [a] -> [a]
- which removes duplicate elements from a list.
The function nub
is defined in the standard library module
Data.List
:
you must write import Data.List
at the beginning of
any Haskell program that uses it.
If you do not have time to do all these exercises, don't worry.
The exercises are intended to provide enough work to keep the most
experienced students busy. If you do all exercises marked with an (*) you have
probably understood this week's material.
Good luck!
0 (*). Defining Functions over Lists
(Based on Thompson's book, Chapter 7)
A. The prelude defines a function take
which is
used to take a given number of elements from a list. For example,
take 5 "Programming in Haskell is fun!" = "Progr"
A possible implementation of take
is
take :: Int -> [a] -> [a]
take 0 _ = []
take _ [] = []
take n (x:xs) | n > 0 = x : take (n-1) xs
take _ _ = error "PreludeList.take: negative argument"
Use this definition of take
as a guide to
implement the prelude functions drop
and splitAt
.
B. How would you define a function zip3
which zips
together three lists? Try to write a recursive definition and also
one which uses zip
instead; what are the advantages and
disadvantages of the two definitions?
1. Permutations
A permutation of a list is another list with the same elements,
but in a possibly different order. For example, [1,2,1] is a permutation
of [2,1,1], but not of [1,2,2].
Write a function
isPermutation :: Eq a => [a] -> [a] -> Bool
that returns True
if its arguments are permutations of each other.
Express suitable properties of the reverse
function in the context of permutations.
2. Avoiding Duplicates
(repeated from last week)
In many situations, lists should not contain duplicate elements. For
example, a pack of cards should not contain the same card twice. Define a
function
duplicates :: Eq a => [a] -> Bool
which returns True
if its argument contains duplicate elements.
duplicates [1,2,3,4,5]
False
duplicates [1,2,3,2]
True
Hint: the standard function elem
, which tests whether
an element occurs in a list, is helpful here.
One way to ensure a list contains no duplicates is to start with a
list that might contain duplicate elements, and remove them.
Define a function
removeDuplicates :: Eq a => [a] -> [a]
which returns a list containing the same elements as its argument, but
without duplicates. Test it using the following property:
prop_duplicatesRemoved :: [Integer] -> Bool
prop_duplicatesRemoved xs = not (duplicates (removeDuplicates xs))
Does this property guarantee that removeDuplicates behaves correctly?
If not, what is missing?
(removeDuplicates
is actually a standard function,
called nub
).
3. Pascal's Triangle
Pascal's triangle is a triangle of numbers
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
.............
computed as follows:
- The first row just contains a 1.
- The following rows are computed by adding together adjacent numbers in
the row above, and adding a 1 at the beginning and at the end.
Pascal's triangle is related to the binomial theorem.
Define a function
pascal :: Int -> [Int]
so that pascal
n computes the nth
row of Pascal's triangle.
4. Erastosthenes' sieve
Eratosthenes' sieve is an ancient method for finding prime numbers. Start by
writing out all the numbers from 2 to (say) 100. The first number (2) is
prime. Now cross out all multiples of 2. The first remaining number (3) is
also prime. Cross out all multiples of 3. The first remaining number (5) is
also prime... and so on. When no numbers remain, you have found all the prime
numbers in the range you started with.
Define a function
crossOut :: Int -> [Int] -> [Int]
so that crossOut
m ns
removes all multiples of m from ns. Try to not
implement crossOut
recursively,
but use a list comprehension instead!
Now define a (recursive!) function
sieve :: [Int] -> [Int]
which applies Eratosthenes' sieve to the list of numbers it is given, and
returns a list of all the prime numbers that it found. This is a recursive
function with a list as its argument, so you must see to it that the
list gets smaller in each recursive call. Take an empty argument list as your
base case.
Use sieve to construct the list of primes from 2 to 100.
5. Number Games
In these examples we'll investigate the properties of prime numbers in the
range 2 to 100. Define functions
- To test whether n is a prime (in the range 2 to 100).
- To test whether n is a sum of two primes (in the range 2 to 100).
It is hypothesized that every even number greater than two can be expressed as
the sum of two primes. For example, 4 = 2+2, 6 = 3+3, 8 = 3+5. Is this true
for all even numbers in the range 4 to 100?
6 (*). Occurrences in Lists
Define the following functions, and state their (polymorphic) types:
occursIn x xs
,
which returns True
if x
is an element
of xs
.
allOccurIn xs ys
,
which returns True
if all of the elements of xs
are also elements of ys
.
sameElements xs ys
,
which returns True
if xs
and ys
have exactly the same elements.
numOccurrences x xs
,
which returns the number of times x
occurs in xs
.
In the implementations of the above functions, try to not use recursion, but
use a list comprehension instead!
In some ways, lists are like sets: both are collections of elements. But the
order of elements in a list matters, while it does not matter in a set, and
the number of occurrences in a list matters, while it does not matter in a
set.
The concept of a bag is something between a list and a set: the
number of occurrences matters, but the order of elements does not. One way to
represent a bag is a list of pairs of values and the number of times the value
occurs: for example
[("a",1), ("b",2)]
Define a function bag
to convert a list into a bag. For example,
bag "hello"
should be
[('h',1),('e',1),('l',2),('o',1)]
7 Elements and Positions
Elements which occur in lists do so at a particular position. For example, 'l'
occurs in "hello" at positions 3 and 4. Define functions
positions xs
,
which converts a list into a list of pairs of elements and
their positions. Hint: Make use of the standard function zip
.
firstPosition x xs
,
which returns the first position at which x occurs in xs.
remove1 x xs
,
which removes the first occurrence of x from xs. For
example, remove1 'l' "hello" == "helo"
remove n x xs
,
which removes the first n occurrences of x from xs.
8 (*). More List Comprehensions
Experiment with the function
pairs :: [a] -> [b] -> [(a,b)]
pairs xs ys = [(x,y) | x<-xs, y<-ys]
and see what it does.
A Pythagorean triad is a triple of integers
(a,b,c) such that
a2 + b2 = c2.
Find all Pythagorean triads with a≤b≤c≤100.