Question

In set theory, we know that the distributive formulas hold: A∩(B∪C)=(A∩B)∪(A∩C),A∪(B∩C)=(A∪B)∩(A∪C) Now that we learned formal logic, we can show that these formulas are purely formal in the following sense: For each of them, find an argument in predicate logic and show that it is valid. Then give an interpretation, so that the distributive formulas in set theory follows directly.

Answer 1

In set theory, the distributive formulas A∩(B∪C)=(A∩B)∪(A∩C) and A∪(B∩C)=(A∪B)∩(A∪C) can be proven to be valid arguments in predicate logic. An interpretation using sets of numbers demonstrates that the formulas hold true.

Step-by-step explanation:

The provided formulas in set theory, A∩(B∪C)=(A∩B)∪(A∩C) and A∪(B∩C)=(A∪B)∩(A∪C), can be proven to be valid arguments in predicate logic.

Let's take the first formula, A∩(B∪C)=(A∩B)∪(A∩C), and create an argument:

1. Premise 1: x∈A and x∈(B∪C) (Assume x is an element in both sets A and (B∪C))
2. Premise 2: x∈B or x∈C (Definition of union)
3. Therefore, x∈A∩B or x∈A∩C (Split x and apply the definition of intersection to both premises)
4. Conclusion: x∈(A∩B)∪(A∩C) (Combine the two conclusions from premise 3)

This argument shows that if the premises are true, then the conclusion must be true, proving the validity of the formula.

To give an interpretation that directly relates to the distributive formulas in set theory, let's consider the sets A, B, and C as sets of numbers. For example:

• A = {2, 4, 6, 8}
• B = {4, 6, 8, 10}
• C = {6, 8, 10, 12}

By substituting these numbers into the formulas, we can see that they hold true:

• A∩(B∪C) = {4, 6, 8} = (A∩B)∪(A∩C)
• A∪(B∩C) = {2, 4, 6, 8, 10, 12} = (A∪B)∩(A∪C)