Let A, B and C be the sets such that A ∪ B = A ∪ C and A ∩ B = A ∩ C. Show that B = C.
To prove that B = C using the given information, we'll utilize basic set theory concepts and logical deductions. Let's break down the steps:
1. Start with the given equation A ∪ B = A ∪ C. This equation states that the union of sets A and B is equal to the union of sets A and C.
2. Recall the set union property: A ∪ B = B ∪ A, which means that the order of the sets in a union operation doesn't matter. We can rewrite the equation as B ∪ A = A ∪ C.
3. In the equation B ∪ A = A ∪ C, we can apply the transitive property of equality, which states that if A = B and B = C, then A = C. This allows us to derive B ∪ A = A ∪ C ⟹ B ∪ A = A ∪ B.
4. Using the distributive property of set intersection over union, we can transform B ∪ A = A ∪ B into (B ∪ A) ∩ (A ∪ B) = B ∩ (A ∪ B). The left-hand side of the equation represents the intersection of two sets, while the right-hand side is the intersection of set B with the union of sets A and B.
5. Simplify the equation (B ∪ A) ∩ (A ∪ B) = B ∩ (A ∪ B) by applying the set identity law, which states that A ∩ (A ∪ B) = A. Thus, we get B ∪ A = B.
6. Explore the equation B ∪ A = B further. We know that A ∪ B = B, so substituting B for A ∪ B, we obtain B ∪ B = B.
7. Apply the idempotent law of sets, which states that for any set A, A ∪ A = A. By this law, B ∪ B reduces to B, yielding B = B.
8. From step 7, we conclude that B = B. Applying the reflexive property of equality, which says anything is equal to itself, we find that B = C.
Therefore, the equation A ∪ B = A ∪ C and A ∩ B = A ∩ C implies that B = C.