Suppose A, B, and C are sets. In at least one of the following situations, one can conclude that A = B.
Identify those cases where this conclusion does apply.
(i) A ∪ C = B ∪ C
A={1,2},A={1,2}, B={2,3}B={2,3} and C={1,2,3}.C={1,2,3}. It is A∪B=A∪C=C={1,2,3}.A∪B=A∪C=C={1,2,3}. However, A≠B.
(ii) A ∩ C = B ∩ C
={1,2},A={1,2}, B={2,3}B={2,3} and C={2}.C={2}. It is A∩B=A∩C=C={2}.A∩B=A∩C=C={2}. But, A≠B.
(iii) A ∪ C = B ∪ C and A ∩ C = B ∩ C
Need help with (iii)
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To determine whether the conclusion A = B can be made in situation (iii), where A ∪ C = B ∪ C and A ∩ C = B ∩ C, we need to analyze the conditions and compare the sets A and B.
Given:
A ∪ C = B ∪ C
A ∩ C = B ∩ C
Let's break down the conditions:
1. A ∪ C = B ∪ C:
This condition implies that the union of sets A and C is equal to the union of sets B and C. It means that any element that is in A or C must also be in B or C, and vice versa. However, this condition does not provide conclusive evidence that A = B, as it is possible for A and B to have some elements that are unique to each set.
2. A ∩ C = B ∩ C:
This condition states that the intersection of sets A and C is equal to the intersection of sets B and C. It means that any element that is common to both A and C is also common to both B and C. However, this condition also does not provide conclusive evidence that A = B, as there may be other elements present in A and B that are not shared with C.
Therefore, based on the conditions provided in (iii), we cannot conclude that A = B. The conditions only guarantee that A and B share the same elements as C. To determine if A = B, we would need additional information or conditions.