Question

Доказать (A\B)объединение(B\A)=(AобъединениеB)\(AпересекаетB)

Answer 1

Всё привильно
 (A\B)объединение(B\A)=(AобъединениеB)\(AпересекаетB)

Answer 2

Proof: (A\B) ∪ (B\A) = (A ∪ B) \ (A ∩ B)

To prove the equality (A\B) ∪ (B\A) = (A ∪ B) \ (A ∩ B), we need to show that both sets contain the same elements.

Let's start by considering the left-hand side of the equation, (A\B) ∪ (B\A):

- (A\B) represents the set of elements that are in A but not in B. - (B\A) represents the set of elements that are in B but not in A. - ∪ denotes the union of two sets, which means combining all the elements from both sets without duplication.

Therefore, (A\B) ∪ (B\A) represents the set of elements that are either in A but not in B or in B but not in A.

Now, let's consider the right-hand side of the equation, (A ∪ B) \ (A ∩ B):

- (A ∪ B) represents the set of elements that are in either A or B or both. - (A ∩ B) represents the set of elements that are common to both A and B. - \ denotes the set difference operation, which means removing the elements that are in the second set from the first set.

Therefore, (A ∪ B) \ (A ∩ B) represents the set of elements that are in either A or B or both, but not in both A and B.

To prove the equality, we need to show that both sets contain the same elements.

Let's consider an element x:

- If x is in (A\B) ∪ (B\A), it means that x is either in A but not in B or in B but not in A. - If x is in (A ∪ B) \ (A ∩ B), it means that x is in either A or B or both, but not in both A and B.

We can see that both cases are equivalent, as an element can only be in either A or B or both, but not in both A and B.

Therefore, we can conclude that (A\B) ∪ (B\A) = (A ∪ B) \ (A ∩ B).

Note: The proof provided above is a logical deduction based on the definitions of set operations. It does not rely on any specific context or domain.