Transitive closure

I want to create a TransitiveClosure() function in python that can input a dictionary and output a new dictionary of the transitive closure.

for example R = {1: [3], 2: [4], 3: [], 4: [1]} will output R R = {1 : [3], 2 : [1, 3, 4], 3 : [], 4 : [1, 3]}.

I know the transitive property is a->b, b->c than a->c.

I can't use a matrix and actually point as I need to create a new dictionary. I've tried converting the dictionary to a list to contain sets but that also has its problems. Could anyone help?

Thank You!

def transitiveClosure(r):
  d ={}
  R = list(r.items())
    # loop for a,b
  for a, b in R:
    #loop for c, d
    for c, d in R:
      # checks if b = c and if a, d are in R
      if b == c and ((a, d) not in R):
        print("Not Transitive")
        return False
  print("is Transitive")
  return True

this will tell me if a dictionary is transitive, I'm having a hard time trying to create a new dictionary using this logic. since I converted the input dictionary into a list of sets do i need to add the elements to a list then convert that back to a dictionary?

I can think of the following solution using a recursive function

def reachable_items(R,k):
    ans = R[k]
    if ans != []:
        for v in ans:
            ans = ans + reachable_items(R,v)
    return ans
    
def transitive_closure(R):
    ans = dict()
    for k in R.keys():
        ans[k] = reachable_items(R,k)
    return ans

Examples:

>>> R1 = {1: [3], 2: [4], 3: [], 4: [1]}
>>> transitive_closure(R1)
{1: [3], 2: [4, 1, 3], 3: [], 4: [1, 3]}

>>> R2 = {1:[2],2:[],3:[4,5],4:[1],5:[]}
>>> transitive_closure(R2)
{1: [2], 2: [], 3: [4, 5, 1, 2], 4: [1, 2], 5: []}