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fp2023:hw1 [2023/03/22 08:57]
pdmatei 		fp2023:hw1 [2023/03/28 14:11] (current)
pdmatei
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 </​code>​ </​code>​
  
-For instance, the set $math[\{1,​2,​3\}] ​may be encoded by the anonymous function:+For instance, the set $math[\{1,​2,​3\}] ​will be encoded by the anonymous function:
 <code scala> <code scala>
 (x: Int) => (x == 1 || x == 2 || x == 3) (x: Int) => (x == 1 || x == 2 || x == 3)
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 </​code>​ </​code>​
  
-**3.** Write a function ''​fromBounds''​ which takes two integer bounds ''​start''​ and ''​stop''​ and returns the set $math[\{start,​ start+1, \ldots, stop\}]. It is guaranteed that ''​start <= stop'' ​(you do not need to check this condition in your implementation(.+**3.** Write a function ''​ins''​ which inserts a new element in a set. More precisely, given $math[x] and $math[set], ''​ins''​ returns a new set $math[\{x\} \cup set]. 
 +<code scala> 
 +def ins(x: Int)(set: Set): Set = ??? 
 +</​code>​ 
 + 
 +**4.** Write a function ''​fromBounds''​ which takes two integer bounds ''​start''​ and ''​stop''​ and returns the set $math[\{start,​ start+1, \ldots, stop\}]. It is guaranteed that $math[start \leq stop] (you do not need to check this condition in your implementation).
  
 <code scala> <code scala>
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 </​code>​ </​code>​
  
-**4.** Write the function which performs the union of two sets:+**5.** Write the function which performs the union of two sets:
 <code scala> <code scala>
 def union(set1: Set, set2: Set): Set = ??? def union(set1: Set, set2: Set): Set = ???
 </​code>​ </​code>​
  
-**5.** Write a function which computes the complement of a set with respect to the set of integers:+**6.** Write a function which computes the complement of a set with respect to the set of integers:
 <code scala> <code scala>
 def complement(s1:​ Set): Set = ??? def complement(s1:​ Set): Set = ???
 </​code>​ </​code>​
  
-**6.** Write a function which computes the sum of value ''​b''​ to all elements from a set, for given **bounds**. Use a tail-end recursive function:+**7.** Write a function which computes the sum of value ''​b''​ to all elements from a set, for given **bounds**. Use a tail-end recursive function:
 <code scala> <code scala>
   def sumSet(b: Int)(start: Int, stop: Int)(set: Set): Int = {   def sumSet(b: Int)(start: Int, stop: Int)(set: Set): Int = {
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 </​code>​ </​code>​
  
-**7.** Generalise the previous function such that we can **fold** a set using any binary commutative operation over integers. Make sure this is a **left** fold: Folding the set: ''​{x,​y,​z}''​ with ''​b''​ should produce: ''​( (b op x) op y) op z''​+**8.** Generalise the previous function such that we can **fold** a set using any binary commutative operation over integers. Make sure this is a **left** fold: Folding the set: ''​{x,​y,​z}''​ with ''​b''​ should produce: ''​( (b op x) op y) op z''​
 <code scala> <code scala>
   def foldLeftSet   def foldLeftSet
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 </​code>​ </​code>​
  
-**8.** Implement an alternative to the previous function, namely **foldRight**. Applying ''​foldRight''​ on the set ''​{x,​y,​z}''​ with ''​b''​ should produce: ''​a op (b op (c op b))''​. Use direct recursion instead of tail recursion.+**9.** Implement an alternative to the previous function, namely **foldRight**. Applying ''​foldRight''​ on the set ''​{x,​y,​z}''​ with ''​b''​ should produce: ''​a op (b op (c op b))''​. Use direct recursion instead of tail recursion.
 <code scala> <code scala>
   def foldRightSet   def foldRightSet
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 </​code>​ </​code>​
  
-**9.** Implement operation ''​filter''​ which takes a set and returns another one containing only those elements that satisfy the predicate:+**10.** Implement operation ''​filter''​ which takes a set and returns another one containing only those elements that satisfy the predicate:
 <code scala> <code scala>
 def filter(p: Int => Boolean)(set:​ Set): Set = ??? def filter(p: Int => Boolean)(set:​ Set): Set = ???
 </​code>​ </​code>​
  
-**10.** Implement a function which **partitions** a set into two sets. The left-most contains those elements that satisfy the predicate, while the right-most contains those elements that do not satisfy the predicate. Use pairs. A pair is constructed with simple parentheses: ''​(1,​2)''​. Suppose ''​val p: (Int,​Int)''​ is a pair of two integers. Then ''​p._1''​ is the left-most part of the pair while ''​p._2''​ is the right-most part of the pair.+**11.** Implement a function which **partitions** a set into two sets. The left-most contains those elements that satisfy the predicate, while the right-most contains those elements that do not satisfy the predicate. Use pairs. A pair is constructed with simple parentheses. E.g. ''​(1,​2)'' ​is a pair of two integers. Suppose ''​val p: (Int,​Int)''​ is another ​pair of two integers. Then ''​p._1''​ is the left-most part of the pair while ''​p._2''​ is the right-most part of the pair.
 <code scala> <code scala>
   def partition(p:​ Int => Boolean)(set:​ Set): (Set,Set) = ???   def partition(p:​ Int => Boolean)(set:​ Set): (Set,Set) = ???
 </​code>​ </​code>​
  
-**11.** Implement a function ''​forall''​ which checks if all elements in a given range of a set satisfy a predicate (condition). (Such a condition may be that all elements from given bounds are even numbers).+**12.** Implement a function ''​forall''​ which checks if all elements in a given range of a set satisfy a predicate (condition). (Such a condition may be that all elements from given bounds are even numbers).
 <code scala> <code scala>
   def forall(cond:​ Int => Boolean) // condition to be checked   def forall(cond:​ Int => Boolean) // condition to be checked
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 </​code>​ </​code>​
  
-**12.** Implement a function ''​exists''​ which checks if a predicate holds for **some** element from the range of a set. Hint: it is easier to implement ''​exists''​ using the logical relation: $math[ \exists x. P(X) \iff \lnot \forall x.\lnot P(X)].+**13.** Implement a function ''​exists''​ which checks if a predicate holds for **some** element from the range of a set. Hint: it is easier to implement ''​exists''​ using the logical relation: $math[ \exists x. P(X) \iff \lnot \forall x.\lnot P(X)].
  
-**13.** Implement the function ''​setOfDivByK''​ which returns the set of integers divisible by a value ''​k''​. Use the appropriate functions you have defined.+**14.** Implement the function ''​setOfDivByK''​ which returns the set of integers divisible by a value ''​k''​. Use the appropriate functions you have defined.
 <code scala> <code scala>
 def setOfDivByK(k:​ Int): Set = ?? def setOfDivByK(k:​ Int): Set = ??
 </​code>​ </​code>​
  
-**14.** Implement the function ''​moreDivs''​ which verifies if ''​set1''​ contains more divisors of ''​k''​ than ''​set2'',​ over the range ''​[start,​stop]''​. Use any combination of the previous functions you have defined for your implementation.+**15.** Implement the function ''​moreDivs''​ which verifies if ''​set1''​ contains more divisors of ''​k''​ than ''​set2'',​ over the range ''​[start,​stop]''​. Use any combination of the previous functions you have defined for your implementation.
 <code scala> <code scala>
 def moreDivs(k: Int)(start: Int, stop:​Int)(set1:​ Set, set2: Set): Boolean = ??? def moreDivs(k: Int)(start: Int, stop:​Int)(set1:​ Set, set2: Set): Boolean = ???
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   * Please follow the [[fp2023:​submission-guidelines| Submission guidelines]] which are the same for all homework.   * Please follow the [[fp2023:​submission-guidelines| Submission guidelines]] which are the same for all homework.
-  * To solve your homework, download the {{|Project template}}, import it in IntellIJ, and you are all set. Do not rename the project manually, as this may cause problems with IntellIJ.+  * To solve your homework, download the {{:​fp2023:​hw1-functions-as-sets.zip|Project template}}, import it in IntellIJ, and you are all set. Do not rename the project manually, as this may cause problems with IntellIJ.