Set Theory and Truth Table Exercises

Set Theory Exercises in Python

Exercise 1: Basic Set Operations

# Define some sets
A = {1, 2, 3, 4, 5}
B = {4, 5, 6, 7, 8}
C = {2, 4, 6, 8, 10}

# Union (A ∪ B)
union_AB = A.union(B)
print(f"A ∪ B = {union_AB}")

# Intersection (A ∩ B)
intersection_AB = A.intersection(B)
print(f"A ∩ B = {intersection_AB}")

# Difference (A - B)
difference_AB = A.difference(B)
print(f"A - B = {difference_AB}")

# Symmetric difference (A Δ B)
sym_diff_AB = A.symmetric_difference(B)
print(f"A Δ B = {sym_diff_AB}")

# Subset check
print(f"Is 4 ⊆ A? {set([4, 5]).issubset(A)}")

Output:

A ∪ B = {1, 2, 3, 4, 5, 6, 7, 8}
A ∩ B = {4, 5}
A - B = {1, 2, 3}
A Δ B = {1, 2, 3, 6, 7, 8}
Is {4, 5} ⊆ A? True

Exercise 2: Set Relationships and Logic

# Knowledge representation using sets
animals = {"cat", "dog", "bird", "fish"}
mammals = {"cat", "dog", "whale", "bat"}
pets = {"cat", "dog", "bird", "hamster"}

# Which animals are both mammals and pets?
mammal_pets = mammals.intersection(pets)
print(f"Mammal pets: {mammal_pets}")

# Which pets are not mammals?
non_mammal_pets = pets.difference(mammals)
print(f"Non-mammal pets: {non_mammal_pets}")

# Are all pets animals? (subset check)
all_pets_are_animals = pets.issubset(animals)
print(f"All pets are animals: {all_pets_are_animals}")

Output:

Mammal pets: {'cat', 'dog'}
Non-mammal pets: {'bird', 'hamster'}
All pets are animals: False

Truth Table Exercise

Boolean Logic Mapping to Set Operations

Set Operation	Boolean Logic	Truth Table
A ∪ B	A OR B	T when either A or B is true
A ∩ B	A AND B	T when both A and B are true
A’	NOT A	T when A is false

Example Truth Table: (A ∩ B) ∪ C

def truth_table_example(A, B, C):
    """Demonstrate (A ∩ B) ∪ C using sets and boolean logic"""
    
    # Set representation
    result_set = (A.intersection(B)).union(C)
    
    # Boolean logic equivalent
    # For each element in universal set, check if it satisfies the condition
    universal = A.union(B).union(C)
    result_bool = set()
    
    for x in universal:
        if (x in A and x in B) or (x in C):
            result_bool.add(x)
    
    print(f"Set result: {result_set}")
    print(f"Boolean result: {result_bool}")
    print(f"Results match: {result_set == result_bool}")

# Test the function
A = {1, 2, 3}
B = {2, 3, 4}
C = {3, 4, 5}

truth_table_example(A, B, C)

Output:

Set result: {2, 3, 4, 5}
Boolean result: {2, 3, 4, 5}
Results match: True

Key Insights

1. Set operations directly map to logical operations - Union is OR, Intersection is AND
2. Truth tables help visualize when logical expressions are true or false
3. Python sets make it easy to experiment with these concepts
4. Knowledge representation often uses these fundamental operations to define relationships

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