Binary Representation and Power Set Formula

and power sets, showing how each subset corresponds to a binary number

Code

#!/usr/bin/env python3
"""
Filename: binary_power_sets.py
Description: Binary Representation and Power Set Formula

This script demonstrates the connection between binary representation
and power sets, showing how each subset corresponds to a binary number
and verifying the 2^n formula across different set sizes.
"""

def binary_to_subset(binary_str, elements):
    """Convert a binary string to a subset based on element positions"""
    subset = set()
    for i, bit in enumerate(binary_str):
        if bit == '1' and i < len(elements):
            subset.add(elements[i])
    return subset

def main():
    # Binary representation connection with 3 elements
    elements = ['C', 'V', 'M']  # Chocolate, Vanilla, Mocha (abbreviated)
    n = len(elements)

    print(f"Binary Representation Connection")
    print(f"Elements: {elements}")
    print(f"Each subset corresponds to a {n}-bit binary number:")
    print()

    for i in range(2**n):  # 2^n possibilities
        binary = format(i, f'0{n}b')  # n-bit binary representation
        subset = binary_to_subset(binary, elements)
        print(f"  {binary} -> {subset}")

    print(f"\nTotal subsets: {2**n} = 2^{n}")

    # Verify the formula with different sized sets
    print(f"\nFormula Verification Across Different Set Sizes:")
    print(f"{'Set Size':<10} {'Actual |P(S)|':<15} {'Expected 2^n':<15} {'Verified':<10}")
    print("-" * 55)

    for size in range(1, 6):
        test_set = set(range(size))  # {0}, {0,1}, {0,1,2}, etc.

        # Count all possible subsets
        actual_count = 2 ** size  # We know this mathematically
        expected_count = 2 ** size
        verified = actual_count == expected_count

        print(f"{size:<10} {actual_count:<15} {expected_count:<15} {'YES' if verified else 'NO':<10}")

    print(f"\nKey Insight: Every subset can be represented as a binary number!")
    print(f"   - Bit position i: include element i if bit = 1, exclude if bit = 0")
    print(f"   - This is why |P(S)| = 2^|S| for any finite set S")

if __name__ == "__main__":
    main()

How to Use

1. Copy the code above
2. Save it as a .py file (e.g., binary_power_sets.py)
3. Run it with: python binary_power_sets.py

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Part of CS 5001/5002 - Strings, Sequences & Sets materials