Question

Let A, B, and C be sets. Prove that A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) using formal definitions.

Answer 1

To prove that A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C), we need to show that both sets contain the same elements. We can do this by going through each element in each set and showing that it belongs to the other set as well. This establishes the equality of the two sets.

Step-by-step explanation:

To prove that A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C), we need to show that both sets contain the same elements. Let's start by proving that every element in A ∪ (B ∩ C) is also in (A ∪ B) ∩ (A ∪ C).

1. Take an element x that belongs to A ∪ (B ∩ C).
2. This means that x is either in set A or in the intersection of sets B and C.
3. If x is in A, then it is in A ∪ B and A ∪ C.
4. If x is in the intersection of sets B and C, then it is in B and C individually, and hence it is in A ∪ B and A ∪ C.
5. In both cases, we can conclude that x is in (A ∪ B) ∩ (A ∪ C).

Next, we need to prove that every element in (A ∪ B) ∩ (A ∪ C) is also in A ∪ (B ∩ C).

1. Take an element y that belongs to (A ∪ B) ∩ (A ∪ C).
2. This means that y is in both A ∪ B and A ∪ C.
3. If y is in A, then it is also in A ∪ (B ∩ C).
4. If y is in B and not in A, it cannot be in A ∪ (B ∩ C) because B ∩ C is a subset of B and not A. Therefore, y must be in A ∪ (B ∩ C).

Since we have shown that every element in A ∪ (B ∩ C) is in (A ∪ B) ∩ (A ∪ C) and vice versa, we can conclude that A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C).