from itertools import combinations

# Step 1: Define the set and target sum
numbers = list(range(1, 9))
target = 9

# Step 2: Generate all 5-number combinations
valid_combinations = list(combinations(numbers, 5))

# Step 3: Define the 4 mutually exclusive pairs that sum to 9
pairs = [(1, 8), (2, 7), (3, 6), (4, 5)]

# Step 4: Check each combination for the required condition
def has_pair_summing_to_target(combo, pair_list):
    for a, b in pair_list:
        if a in combo and b in combo:
            return True
    return False

# Step 5: Verify each combination
always_has_pair = all(has_pair_summing_to_target(c, pairs) for c in valid_combinations)

print(f"Every 5-number combination has at least one pair summing to {target}: {always_has_pair}")

'''
The code verifies if every 5-number combination from 1 to 8 contains at least one of the specified pairs that sum to 9.
# Output: True or False based on the verification
# Note: The output will be True since every combination of 5 numbers from 1 to 8 will always
#  include at least one of the pairs (1, 8), (2, 7), (3, 6), or (4, 5) due to the nature of combinations and the range of numbers.
How it works:
1. It generates all combinations of 5 numbers from the set {1, 2, 3, 4, 5, 6, 7, 8}.
2. It checks each combination to see if it contains at least one of the pairs that sum to 9.        
'''