type Element = (Int,Int)
type BinaryRelation = Element -> Bool
bound = 99

-- returns an empty binary relation, i.e. no pairs
emptyBinaryRelation :: BinaryRelation
???

-- returns the universal binary relation, i.e. all pairs
universalBinaryRelation :: BinaryRelation
???

-- returns True of element is in binary relation; False otherwise
contains :: Element -> BinaryRelation -> Bool
???

-- returns a new binary relation obtained by adding element to binary relation; 
-- if element is already in binary relation then returns same binary relation
add :: Element -> BinaryRelation -> BinaryRelation
???

-- returns new binary relation obtained by adding all elements in input list to
-- given binary relation.
addMultiple :: [Element] -> BinaryRelation -> BinaryRelation
???

-- returns True if binary relation 1 is a subset of binary relation 2;
-- False otherwise.
subRelation :: BinaryRelation -> BinaryRelation -> Bool
subRelation r1 r2 = subRelationHelper1 r1 r2 0

subRelationHelper1 :: BinaryRelation -> BinaryRelation -> Int -> Bool
subRelationHelper1 r1 r2 n 
  | n > bound = True
  | otherwise = subRelationHelper2 r1 r2 n 0 && subRelationHelper1 r1 r2 (n+1)

subRelationHelper2 :: BinaryRelation -> BinaryRelation -> Int -> Int-> Bool
subRelationHelper2 r1 r2 n m 
  | m > bound                                    = True
  | contains (n,m) r1 && not (contains (n,m) r2) = False
  | otherwise                                    = subRelationHelper2 r1 r2 n (m+1)

-- returns True if binary relation 1 and binary relation 2 are equal;
-- False otherwise.
equal :: BinaryRelation -> BinaryRelation -> Bool
???

-- for all a, (a,a) is in binary relation
reflexive :: BinaryRelation -> Bool
???

-- returns the union of the two input binary relations
union :: BinaryRelation -> BinaryRelation -> BinaryRelation
???

-- returns inverse of input binary relation; 
-- inverse(r) = { (b,a) | (a,b) in r }
inverse :: BinaryRelation -> BinaryRelation
???

-- returns True if input binary relation is symmetric; False otherwise
symmetric :: BinaryRelation -> Bool
???

-- returns True if input binary relation is anti-symmetric; False otherwise
antiSymmetric :: BinaryRelation -> Bool
???

-- returns True if input binary relation is transitive; False otherwise
transitive :: BinaryRelation -> Bool
???

-- returns True if input binary relation is an equivalence; False otherwise
equivalence :: BinaryRelation -> Bool
equivalence r = reflexive r && symmetric r && transitive r

-- look up definition in BinaryRelationNotes.pdf
reflexiveClosure :: BinaryRelation -> BinaryRelation
???

-- look up definition in BinaryRelationNotes.pdf
symmetricClosure :: BinaryRelation -> BinaryRelation
???

-- selfJoin will be useful in implementing transitiveClosure;
-- returns selfJoin(r) = { (a,c) | (a,b) is in r and (b,c) is in r }
selfJoin :: BinaryRelation -> BinaryRelation
???

-- look up definition in BinaryRelationNotes.pdf
transitiveClosure :: BinaryRelation -> BinaryRelation
???

-- returns String version of binary relation
toString :: BinaryRelation -> String
toString r = show [(x,y) | x <- [0..bound], y <- [0..bound], r (x,y)]