Question

高一集合问题，求详细解题过程，在线等，急！！！

A∩B=A∩C且AUB=AUC等价于B=C 试证明之

请详细讲解一下！谢！

Answer 1

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) #