FP Cvičení 2

Types

• V interpretu GHCi lze použít příkaz :info k zjištění typu funkce nebo operátoru, zjistěte typ následujících elementů: +, sqrt, succ, max
• Zjistěte informace o typech následujících výrazů a vyhodnoťte je. K tomuto slouží příkaz :type. Je možné zapnout, aby interpret standardně vypsal typ výrazu. Toto jde příkazem :set +t (tuto funkci lze vypnout :unset +t).

5 + 8 
3 * 5 + 8
2 + 4
sqrt 16 
succ 6
succ 7
pred 9
pred 8
sin (pi / 2)
truncate pi
round 3.5
round 3.4 
floor 3.7 
ceiling 3.3
mod 10 3
odd 3

• At presentations, we have spoken about some basic types: Int, Double, Bool, Char. For each of previous expressions assign them the most appropriate of these basic data types. You can verify your guess by using ::. For example, for the first expression, let's assume it is Int. We can cast the result to integer and get the following result.

Prelude> :type (5 + 8) :: Int
(5 + 8) :: Int :: Int

If we try incorrect conversion to Char, we get the following result.

Prelude> :type (5 + 8) :: Char

<interactive>:1:2: error:
    * No instance for (Num Char) arising from a use of `+'
    * In the expression: (5 + 8) :: Char

For this expression, also the type Double works.

Prelude> :type (5 + 8) :: Double
(5 + 8) :: Double :: Double

Simple functions

Implement following functions:

• Function that computes a factorial of a given number.

factorial :: Int -> Int

• Function that computes n-th number in Fibonacci sequence.

fib :: Int -> Int

• Function that checks if a year is a leap-year (divisible without remainder by 4 and it is not divisible by 100. If it is divisible by 400, it is a leap-year).

leapYear :: Int -> Bool

• Implement two functions that returns a maximum from 2 respectively 3 given parameters.

max2 :: Int -> Int -> Int
max3 :: Int -> Int -> Int -> Int

• Term combination is a selection of items from a collection, such that (unlike permutations) the order of elements in this selection does not matter. Compute the number of possible combinations if we are taking k things from the collection of n things.

combinations :: Int -> Int -> Int

• Implement a function that computes the number of solutions for a quadratic equation. This quadratic equation will be given using standard coefficients: a, b, c.

numberOfRoots :: Int -> Int -> Int -> Int
-- To simplify the solution, let construct can be used
f x y = let a = x + y
        in a * a

• Implement a function that computes greatest common divider for two given numbers.

gcd' :: Int -> Int -> Int

• Implement a function that compute, if a given number is a prime number.

isPrime :: Int -> Bool