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Here are some exercises designed to help you practice programming with lists and list comprehensions.
You may need the following useful standard functions:
import Listat 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.
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.
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.
Main> duplicates [1,2,3,4,5]
Main> duplicates [1,2,3,2]
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).
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 .............computed as follows:
Define a function
pascal :: Int -> [Int]so that pascal n computes the nth row of Pascal's triangle.
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.
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
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
a^2 + b^2 == c^2Find all Pythagorean triads with a<=b<=c<=100.