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Show that A intersection B is equal to A intersection C need not imply B is equal to C?
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Show that A intersection B is equal to A intersection C need not imply...
Explanation:
Given: A ∩ B = A ∩ C
Counterexample: B ≠ C
- Let's consider a counterexample to show that A ∩ B = A ∩ C does not imply B = C.
- Suppose A = {1, 2, 3}, B = {1, 2}, and C = {1, 2}.
- Then A ∩ B = {1, 2} and A ∩ C = {1, 2}.
- Here, A ∩ B = A ∩ C, but B ≠ C since B = {1, 2} and C = {1, 2} are not equal.
Explanation:
- The intersection of sets A and B being equal to the intersection of sets A and C does not necessarily mean that sets B and C are equal.
- When two sets have the same intersection with a third set, it means that they share some common elements with the third set.
- However, this does not guarantee that the two sets are the same, as they could have additional elements that are not shared with the third set.
- The equality of intersections only implies that the sets B and C have at least the same elements in common with set A, but they can still have distinct elements that differentiate them from each other.
Given: A ∩ B = A ∩ C
Counterexample: B ≠ C
- Let's consider a counterexample to show that A ∩ B = A ∩ C does not imply B = C.
- Suppose A = {1, 2, 3}, B = {1, 2}, and C = {1, 2}.
- Then A ∩ B = {1, 2} and A ∩ C = {1, 2}.
- Here, A ∩ B = A ∩ C, but B ≠ C since B = {1, 2} and C = {1, 2} are not equal.
Explanation:
- The intersection of sets A and B being equal to the intersection of sets A and C does not necessarily mean that sets B and C are equal.
- When two sets have the same intersection with a third set, it means that they share some common elements with the third set.
- However, this does not guarantee that the two sets are the same, as they could have additional elements that are not shared with the third set.
- The equality of intersections only implies that the sets B and C have at least the same elements in common with set A, but they can still have distinct elements that differentiate them from each other.
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Show that A intersection B is equal to A intersection C need not imply B is equal to C?
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Show that A intersection B is equal to A intersection C need not imply B is equal to C? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Show that A intersection B is equal to A intersection C need not imply B is equal to C? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Show that A intersection B is equal to A intersection C need not imply B is equal to C?.
Show that A intersection B is equal to A intersection C need not imply B is equal to C? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Show that A intersection B is equal to A intersection C need not imply B is equal to C? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Show that A intersection B is equal to A intersection C need not imply B is equal to C?.
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