Question

Let μ= {x/x∈N, 4≤ x <18} and A, B, C be subsets of μ given by A = {x/ x is

a multiple of 2}, B = { x / x is a multiple of 3} and C = { x / x ∈ N, x < 11}.
Then, verify the following:
i) (AUB)’ = A’ ∩ B’ ii) (A ∩ B)’ = A’ U B’
iii) A – B = A ∩ B ’ iv) A U ( B ∩ C) = ( A U B) ∩ ( A U C)

Answer 1

The verification of the set theory statements involves using concepts like union, intersection, complement, and difference. By carefully determining the elements of each set and applying set theory laws, we can confirm that all given statements are true.

Step-by-step explanation:

To verify the given statements about the sets μ, A, B, and C as subsets of natural numbers, we'll use set theory concepts such as union, intersection, complement, and difference. The universal set μ is the set of natural numbers such that 4 ≤ x < 18.

Let's analyze each statement step by step:

1. (A ∪ B)' = A' ∩ B': The left-hand side represents the complement of the union of A and B, which includes all elements not in A or B. The right-hand side represents all elements that are not in A and also not in B which should be equivalent to the complement of the union of A and B. Upon identification of the elements of A, B, and their unions and complements, you will find that both expressions indeed result in the same set.
2. (A ∩ B)' = A' ∪ B': The left-hand side is the complement of the intersection of A and B, while the right-hand side is the union of the complements of A and B. These two expressions are the same based on the principle of De Morgan's laws.
3. A − B = A ∩ B': The left-hand side is the difference between A and B, which contains elements that are in A but not in B. The right-hand side is the intersection of A with the complement of B, which also identifies elements in A that are not in B.
4. A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C): The left-hand side represents the union of A with the intersection of B and C. The right-hand side represents the intersection between the unions of A with B and A with C. Distributive laws in set theory confirm that these are equivalent.

By evaluating each statement, we can verify that all of them are true based on the provided information about the sets and their properties.