Week 8: Exercises

1. Function Types

What are the types of these functions?

1. triple x = 3 * x

2. even x = (x `mod` 2 == 0)

3. palindrome xs = (reverse xs == xs)

4. f x = f x

2. List Functions

Write these functions (which are all built into the standard library):

a) isPrefixOf :: Eq a => [a] → [a] → Bool

Take two lists and return true if the first list is a prefix of the second.

b) isInfixOf :: Eq a => [a] → [a] → Bool

Take two lists and return true if the first list is contained anywhere within the second.

c) group :: Eq a => [a] → [[a]]

Group adjacent identical elements into sublists. For example,

group "Mississippi" == ["M","i","ss","i","ss","i","pp","i"]

3. Cyclic List

Implelement the built-in function cycle that takes a list L and returns an infinite list consisting of L repeated over and over:

> take 10 (cycle "abc")
"abcabcabca"

4. Fibonacci Numbers

Construct an infinite list containing all Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13, ...

5. Prime

Write a function that determines whether an integer is prime.

6. All Primes

Construct an infinite list containing all prime numbers.

7. Greatest Common Divisor

Write a function that computes the greatest common divisor of two numbers. It should work with any integral type, e.g. either Int or Integer.

8. Vector Sum

Write a function add_vec that adds two vectors represented as lists. Give your function the most general possible type.

9. Dot Product

Write a function dot that computes the dot product of two vectors represented as lists.

10. Matrix Addition

Write a function add that adds two matrices represented as lists of lists.

11. All Pairs

Construct an infinite list allPairs that contains all pairs of positive integers. Every pair must appear exactly once in the list.