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Use the properties of sets to prove that for all the sets A and B: A – (A ∩ B) = A – B

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Use the properties of sets to prove that for all the sets A and B:

A – (A ∩ B) = A – B

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We have 

A – (A ∩ B) = A ∩ (A ∩ B)′ (since A – B = A ∩ B′) 

= A ∩ (A′ ∪ B′) [by De Morgan’s law) 

= (A∩A′) ∪ (A∩ B′) [by distributive law] 

= φ ∪ (A ∩ B′) 

= A ∩ B′ = A – B

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