Question:
Show that \(A ∩ B = A ∩ C\) need not imply B = C.
Show that \(A ∩ B = A ∩ C\) need not imply B = C.
Updated On: Jan 21, 2026
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Solution and Explanation
Let A = {0, 1}, B = {0, 2, 3}, and C = {0, 4, 5}
Accordingly, \(A ∩ B = \){0} and \(A ∩ C =\) {0}
Here, \(A ∩ B = A ∩ C = \){0}
However, \(B ≠ C [2 ∈ B \) and \(2 ∉ C]\)
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Concepts Used:
Operations on Sets
Some important operations on sets include union, intersection, difference, and the complement of a set, a brief explanation of operations on sets is as follows:
1. Union of Sets:
- The union of sets lists the elements in set A and set B or the elements in both set A and set B.
- For example, {3,4} ∪ {1, 4} = {1, 3, 4}
- It is denoted as “A U B”
2. Intersection of Sets:
- Intersection of sets lists the common elements in set A and B.
- For example, {3,4} ∪ {1, 4} = {4}
- It is denoted as “A ∩ B”
3.Set Difference:
- Set difference is the list of elements in set A which is not present in set B
- For example, {3,4} - {1, 4} = {3}
- It is denoted as “A - B”
4.Set Complement:
- The set complement is the list of all elements present in the Universal set except the elements present in set A
- It is denoted as “U-A”