Question

What is the value of (A U B U C) ∩ (A ∩ B' ∩ C') ∩ C'?

Answer 1

SOLUTION

TO DETERMINE

(A U B U C) ∩ (A ∩ B' ∩ C')' ∩ C'

EVALUATION

( A U B U C ) ∩ ( A ∩ B' ∩ C' )' ∩ C'

= ( A U B U C ) ∩ ( A' U B U C ) ∩ C' [ By De Morgan's law ]

= ( B U C U A) ∩ ( B U C U A') ∩ C'

= [ ( B U C ) U ( A ∩ A') ] ∩ C' [ By Distributive Property ]

= [ ( B U C ) U Φ ] ∩ C'

= ( B U C ) ∩ C'

= ( B ∩ C' ) U ( C ∩ C' ) [ By Distributive Property ]

= ( B ∩ C' ) U Φ

= ( B ∩ C' )

= B - C

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A∩(BUC) = (A∩B) U (A∩C).

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Answer 2

Answer:

TO PROVE :-

(A U B U C) ∩ (A ∩ B' ∩ C')' ∩ C'

PROOF :-

( A U B U C ) ∩ ( A ∩ B' ∩ C' )' ∩ C'

= ( A U B U C ) ∩ ( A' U B U C ) ∩ C' [ By De Morgan's law ]

= ( B U C U A) ∩ ( B U C U A') ∩ C'

= [ ( B U C ) U ( A ∩ A') ] ∩ C' [ By Distributive Property ]

= [ ( B U C ) U Φ ] ∩ C'

= ( B U C ) ∩ C'

= ( B ∩ C' ) U ( C ∩ C' ) [ By Distributive Property ]

= ( B ∩ C' ) U Φ

= ( B ∩ C' )

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