Question

Let a and b be sets. prove that a ∪ b = a ∩ b if and only if a = b.

Answer 1

To prove that a ∪ b = a ∩ b if and only if a = b, we need to demonstrate both directions of the statement.

Step-by-step explanation:

To prove that a ∪ b = a ∩ b if and only if a = b, we need to demonstrate both directions of the statement.

Direction 1: If a = b, then a ∪ b = a ∩ b.

1. Assume a = b.
2. By definition, a ∪ b is the set that contains all elements that are in either a or b.
3. Since a = b, the set a ∪ b will also be equal to a (because all elements in b are also in a).
4. Similarly, a ∩ b is the set that contains all elements that are in both a and b.
5. Since a = b, the set a ∩ b will also be equal to a.
6. Therefore, if a = b, then a ∪ b = a ∩ b.

Direction 2: If a ∪ b = a ∩ b, then a = b.

1. Assume a ∪ b = a ∩ b.
2. By definition, a ∪ b is the set that contains all elements that are in either a or b.
3. Similarly, a ∩ b is the set that contains all elements that are in both a and b.
4. Since a ∪ b = a ∩ b, it means that all elements in a ∪ b are also in a ∩ b.
5. Since all elements in a ∪ b are also in a ∩ b, it implies that all elements in a (because a ∪ b contains all elements in a) are also in b (because a ∩ b contains all elements in a).
6. Therefore, if a ∪ b = a ∩ b, then a = b.

Thus, we have proven that a ∪ b = a ∩ b if and only if a = b.