题目
A∩B=A∩C且AUB=AUC等价于B=C 试证明之
提问时间：2021-02-21
答案
A∩B=A∩C and AUB=AUC B=C
if part
if A∩B=A∩C and AUB=AUC
for all x ∈ B
=> x ∈ A∩B
=> x ∈ A∩C (A∩B=A∩C)
=> x ∈ C
=> B is subset of C
for all x ∈ C
=> x ∈ A∩C
=> x ∈ A∩B (A∩B=A∩C)
=> x ∈ B
=> C is subset of B
ie B=C #
only if part
if B=C (trival)
AUB =AUC (B=C)
A∩B=A∩C (B=C) #
if part
if A∩B=A∩C and AUB=AUC
for all x ∈ B
=> x ∈ A∩B
=> x ∈ A∩C (A∩B=A∩C)
=> x ∈ C
=> B is subset of C
for all x ∈ C
=> x ∈ A∩C
=> x ∈ A∩B (A∩B=A∩C)
=> x ∈ B
=> C is subset of B
ie B=C #
only if part
if B=C (trival)
AUB =AUC (B=C)
A∩B=A∩C (B=C) #
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