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If A , B , C be any three sets such that A∪B = A∪C and A∩B=A∩C , then
- a)A = C
- b)A = B
- c)A = B = C
- d)B = C
Correct answer is option 'D'. Can you explain this answer?
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If A , B , C be any three sets such that A∪B = A∪C andA...
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If A , B , C be any three sets such that A∪B = A∪C andA...
Explanation:
Given: AB = AC
We know that AB is the set of elements that are common to both A and B, and AC is the set of elements that are common to both A and C.
Therefore, if AB = AC, it means that the sets B and C have the same elements in common with A.
Now, let's consider the following cases:
1. A is a subset of B and C:
In this case, A is a subset of both B and C, and hence B and C have the same elements as A. Therefore, A = B = C.
2. B is a subset of A and C:
In this case, AB = B, and AC = C. Since AB = AC, we have B = C. Therefore, A, B, and C have the same intersection, and hence B = C is a subset of A.
3. C is a subset of A and B:
In this case, AB = B, and AC = C. Since AB = AC, we have B = C. Therefore, A, B, and C have the same intersection, and hence C = B is a subset of A.
4. A, B, and C have no subset relationship:
In this case, we can consider the Venn diagram of A, B, and C. Since AB = AC, the intersection of A and B is the same as the intersection of A and C. Therefore, the regions outside the intersection are the same for B and C. This means that B and C have the same elements outside the intersection with A. Therefore, B = C is a subset of A.
Hence, in all cases, we have B = C as a subset of A. Therefore, option D is the correct answer.
Given: AB = AC
We know that AB is the set of elements that are common to both A and B, and AC is the set of elements that are common to both A and C.
Therefore, if AB = AC, it means that the sets B and C have the same elements in common with A.
Now, let's consider the following cases:
1. A is a subset of B and C:
In this case, A is a subset of both B and C, and hence B and C have the same elements as A. Therefore, A = B = C.
2. B is a subset of A and C:
In this case, AB = B, and AC = C. Since AB = AC, we have B = C. Therefore, A, B, and C have the same intersection, and hence B = C is a subset of A.
3. C is a subset of A and B:
In this case, AB = B, and AC = C. Since AB = AC, we have B = C. Therefore, A, B, and C have the same intersection, and hence C = B is a subset of A.
4. A, B, and C have no subset relationship:
In this case, we can consider the Venn diagram of A, B, and C. Since AB = AC, the intersection of A and B is the same as the intersection of A and C. Therefore, the regions outside the intersection are the same for B and C. This means that B and C have the same elements outside the intersection with A. Therefore, B = C is a subset of A.
Hence, in all cases, we have B = C as a subset of A. Therefore, option D is the correct answer.
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If A , B , C be any three sets such that A∪B = A∪C andA∩B=A∩C , thena)A = Cb)A = Bc)A = B = Cd)B = CCorrect answer is option 'D'. Can you explain this answer?
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