# question 1 
# create a relation as a set of ordered pairs.
r1 = {(1,2),(2,3),(3,4),(1,3)}
r2 = {(a,b) for a in range(1,4) for b in range (1,4) if a<=b}

print(r1)
print(r2)

# Q2
def is_reflexive(r,elements):
    return all((a,a) in r for a in elements)

# to check if it is antisymetric
def is_antisymmetric(r):
    return all (a==b or (b,a) not in r for (a,b) in r)
# to check if it is transitive
def is_transitive(r):

    return all((a, c) in r for (a, b) in r for (_, c) in r if b == _)
# to check if it is reflexive
def is_partial_order(r, elements):
    
    return is_reflexive(r, elements) and is_antisymmetric(r) and is_transitive(r)

r = {(1, 2), (2, 3), (1, 3)}

# Set of elements
elements = {1, 2, 3}

# Check properties
print("Is the relation reflexive?", is_reflexive(r, elements))

print("Is the relation antisymmetric?", is_antisymmetric(r))

print("Is the relation transitive?", is_transitive(r))
print("Is the relation a partial order?", is_partial_order(r, elements))

#Q3
p=Poset((divisors(24), attrcall("divides")), linear_extension=True)
p.cover_relations()

p.maximal_elements()
p.minimal_elements()
p.show()

 #Q4
P1=Poset((divisors(12),attrcall("divides")),linear_extension=True)
P1.show()
S1=Poset((divisors(6),attrcall("divides")),linear_extension=True)
S1.show()
#upperBound   &   LowerBound
def is_upper_bound(element,S):
    return all(s <= element for s in S)
def is_lower_bound(element,S):
   return all(element <= s for s in S)
is_lower_bound(1,P1)
is_upper_bound(12,P1)
upper_bounds = [u for u in P1 if is_upper_bound(u,S1)]
lower_bounds = [l for l in P1 if is_lower_bound(l,S1)]
lower_bounds
upper_bounds
LUB = min(upper_bounds) if upper_bounds else None
GLB = max(lower_bounds) if lower_bounds else None
LUB 
GLB

#Q5
A1=Poset((divisors(24),attrcall("divides")),linear_extension=True)
A1.show()
A1.is_lattice()

 #Q6
A=Poset({0:[1,2],2:[3,4]})
A.show()
A.is_lattice()