Semigroup

This section refers to Data.Semigroup.

A Semigroup a has function called <>. That function’s type is a -> a -> a, so it takes in two things and puts out one. That’s all. This operation could be concatentation of lists or taking the maximum of two numbers.

You should always use a function that is associative. This means if are going to apply the function twice (so x <> y <> z) it doesn’t matter which operation you do first. Familiar operations like addition and multiplication are associative.

You will see people talking big about this function defining a “binary operation”. That’s a lot of vocabulary to explain that the function takes in two things and puts out one.

1. Write the class declaration for Semigroup a.

2. Define newtype Prod = Prod Int and make Prod an instance of Semigroup by using multiplication.

3. Define data XList b = XList [b] and make (XList b) an instance of Semigroup by appending lists.

4. Define newtype Bite = Bite [String]. Consider making this a semigroup using swapped concatentation:

    instance (Semigroup Bite) where
        (Bite xs) <> (Bite ys) = Bite (ys ++ xs)
   

   1. What is (Bite ["Wolf"]) <> (Bite ["Dog"])?

   2. Is the operation <> associative? Evaluate the code below two different ways as a check. State your conclusion.

       (Bite ["A", "Wolf"]) <> (Bite ["Dancer"]) <> (Bite ["Fang"])
      

5. Define newtype Trouble = Trouble [Int] and consider making this an instance of Semigroup using reversal followed by concatenation:

    instance (Semigroup Trouble) where
        (Trouble xs) <> (Trouble ys) = (reverse xs) ++ (reverse ys)
   

   1. Find (Trouble [1,2,3]) <> (Trouble [4,5,6]).
   2. Is the operation <> associative? Write an example with three lists and evaluate it in two different ways as a check. State your conclusion.