Chapter 1: Sets

Exercise

Question

Question: 8

15 / 17

Show that A ∩ B = A ∩ C need not imply B = C.

Answer

1
Determine what information is provided by the condition A ∩ B = A ∩ C
Venn diagram showing the standard decomposition of two sets into three disjoint regions: A − B, A $\cap$ B, and B − A.
A $∩$ B = A $∩$ C means that the elements common to A and B are the same as the elements common to A and C.
2
Check whether B and C can differ outside their common elements with A.
Answer: Having the same common elements with A does not guarantee that all elements of B and C are the same. Therefore, although the statement may be true in some cases, it is not true in every case.
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