
'''simple docstring'''






import numpy as np
import skfuzzy as fuzz

if __name__ == "__main__":
 # Create universe of discourse in Python using linspace ()
 UpperCAmelCase_       :   Any   =       np.linspace(start=0, stop=75, num=75, endpoint=True, retstep=False)

 # Create two fuzzy sets by defining any membership function
 # (trapmf(), gbellmf(), gaussmf(), etc).
 UpperCAmelCase_       :   Optional[int]   =       [0, 25, 50]
 UpperCAmelCase_       :   Dict   =       [25, 50, 75]
 UpperCAmelCase_       :   Union[str, Any]   =       fuzz.membership.trimf(X, abca)
 UpperCAmelCase_       :   Any   =       fuzz.membership.trimf(X, abca)

 # Compute the different operations using inbuilt functions.
 UpperCAmelCase_       :   int   =       np.ones(75)
 UpperCAmelCase_       :   Tuple   =       np.zeros((75,))
 # 1. Union = max(µA(x), µB(x))
 UpperCAmelCase_       :   str   =       fuzz.fuzzy_or(X, young, X, middle_aged)[1]
 # 2. Intersection = min(µA(x), µB(x))
 UpperCAmelCase_       :   Optional[Any]   =       fuzz.fuzzy_and(X, young, X, middle_aged)[1]
 # 3. Complement (A) = (1- min(µA(x))
 UpperCAmelCase_       :   List[Any]   =       fuzz.fuzzy_not(young)
 # 4. Difference (A/B) = min(µA(x),(1- µB(x)))
 UpperCAmelCase_       :   Union[str, Any]   =       fuzz.fuzzy_and(X, young, X, fuzz.fuzzy_not(middle_aged)[1])[1]
 # 5. Algebraic Sum = [µA(x) + µB(x) – (µA(x) * µB(x))]
 UpperCAmelCase_       :   Dict   =       young + middle_aged - (young * middle_aged)
 # 6. Algebraic Product = (µA(x) * µB(x))
 UpperCAmelCase_       :   List[str]   =       young * middle_aged
 # 7. Bounded Sum = min[1,(µA(x), µB(x))]
 UpperCAmelCase_       :   List[str]   =       fuzz.fuzzy_and(X, one, X, young + middle_aged)[1]
 # 8. Bounded difference = min[0,(µA(x), µB(x))]
 UpperCAmelCase_       :   List[Any]   =       fuzz.fuzzy_or(X, zero, X, young - middle_aged)[1]

 # max-min composition
 # max-product composition

 # Plot each set A, set B and each operation result using plot() and subplot().
 from matplotlib import pyplot as plt

 plt.figure()

 plt.subplot(4, 3, 1)
 plt.plot(X, young)
 plt.title('Young')
 plt.grid(True)

 plt.subplot(4, 3, 2)
 plt.plot(X, middle_aged)
 plt.title('Middle aged')
 plt.grid(True)

 plt.subplot(4, 3, 3)
 plt.plot(X, union)
 plt.title('union')
 plt.grid(True)

 plt.subplot(4, 3, 4)
 plt.plot(X, intersection)
 plt.title('intersection')
 plt.grid(True)

 plt.subplot(4, 3, 5)
 plt.plot(X, complement_a)
 plt.title('complement_a')
 plt.grid(True)

 plt.subplot(4, 3, 6)
 plt.plot(X, difference)
 plt.title('difference a/b')
 plt.grid(True)

 plt.subplot(4, 3, 7)
 plt.plot(X, alg_sum)
 plt.title('alg_sum')
 plt.grid(True)

 plt.subplot(4, 3, 8)
 plt.plot(X, alg_product)
 plt.title('alg_product')
 plt.grid(True)

 plt.subplot(4, 3, 9)
 plt.plot(X, bdd_sum)
 plt.title('bdd_sum')
 plt.grid(True)

 plt.subplot(4, 3, 10)
 plt.plot(X, bdd_difference)
 plt.title('bdd_difference')
 plt.grid(True)

 plt.subplots_adjust(hspace=0.5)
 plt.show()

'''simple docstring'''






import math







def 		A_       (	_lowerCAmelCase   :    int							):



      """simple docstring"""




      _lowerCamelCase				:   Optional[int]								=   math.loga(math.sqrt(4 * positive_integer + 1							) / 2 + 1 / 2							)

      return exponent == int(_lowerCAmelCase							)







def 		A_       (	_lowerCAmelCase   :    float = 1 / 12345							):



      """simple docstring"""




      _lowerCamelCase				:   Optional[int]								=   0
      _lowerCamelCase				:   int								=   0

      _lowerCamelCase				:   List[str]								=   3
      while True:
            _lowerCamelCase				:   List[Any]								=   (integer**2 - 1) / 4
            # if candidate is an integer, then there is a partition for k
            if partition_candidate == int(_lowerCAmelCase							):
                  _lowerCamelCase				:   Any								=   int(_lowerCAmelCase							)
                  total_partitions += 1
                  if check_partition_perfect(_lowerCAmelCase							):
                        perfect_partitions += 1
            if perfect_partitions > 0:
                  if perfect_partitions / total_partitions < max_proportion:
                        return int(_lowerCAmelCase							)
            integer += 1


if __name__ == "__main__":
 print(f'''{solution() = }''')