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Match the following sets for all sets A, B and C

i. ((A′ ∪ B′) – A) ′                     (a) A – B

ii. [B′ ∪ (B′ – A)] ′                    (b) A

iii. (A – B) – (B – C)                 (c) B

iv. (A – B) ∩ (C – B)                 (d) (A × B) ∩ (A × C)

v. A × (B ∩ C)                          (e) (A × B) ∪ (A × C)

vi. A × (B ∪ C)                         (f) (A ∩ C) – B

0 9 Views | Posted 4 months ago
Asked by Shiksha User

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    Answered by

    Vishal Baghel | Contributor-Level 10

    4 months ago

    ( i ) ( ( A ' ∪ B ' ) − A ) ' = [ ( A ' ∪ B ' ) ∩ A ' ] '                                                       [ ? A − B = A ∩ B ' ]                                                                                   = [ ( A ∩ B ) ' ∩ A ' ] '                                                       [ ? A ' ∪ B ' = ( A ∩ B ) ' ]                                                                                   = [ ( A ∩ B ) ' ] ' ∪ ( A ' ) '                                             [ ? ( A ' ) ' = A ]                                                                                   = ( A ∩ B ) ∪ A = A .     S o , ( i )     i s     m a t c h e d     w i t h     ( b ) . ( i i ) [ B ' ∪ ( B ' − A ) ] ' = [ B ' ∪ ( B ' ∩ A ' ) ] '                                                       [ ? A − B = A ∩ B ' ]                                                                                   = ( B ' ) ' ∩ ( B ' ∩ A ' ) '                                                       [ ? A ' ∩ B ' = ( A ∪ B ) ' ]                                                                                   = B ∩ ( B ∪ A ) = B .     S o , ( i i )     i s     m a t c h e d     w i t h     ( c ) . ( i i i ) ( A − B ) − ( B − C ) = ( A ∩ B ' ) − ( B ∩ C ' )                                   [ ? A − B = A ∩ B ' ]                                                                                               = ( A ∩ B ' ) ∩ ( B ∩ C ' ) '                                                                                               = ( A ∩ B ' ) ∩ ( B ' ∪ ( C ' ) ' )                       [ ? A ' ∪ B ' = ( A ∩ B ) ' ]                                                                                               = ( A ∩ B ' ) ∩ ( B ' ∪ C )                                                                                               = [ A ∩ ( B ' ∪ C ) ] ∩ [ B ' ∩ ( B ' ∪ C ) ]                                                                                               = [ A ∩ ( B ' ∪ C ) ] ∩ B '                                                                                               = ( A ∩ B ' ) ∩ ( B ' ∪ C ) ∩ B '                                                                                               = ( A ∩ B ' ) ∩ B ' = ( A ∩ B ' ) = A − B                       S o , ( i i i )     i s     m a t c h e d     w i t h     ( a ) . ( i v ) ( A − B ) ∩ ( C − B ) = ( A ∩ B ' ) ∩ ( C ∩ B ' )                                                       [ ? A − B = A ∩ B ' ]                                                                                               = ( A ∩ C ) ∩ B ' = ( A ∩ C ) − B                       [ ? A ∩ B ' = A − B ]                       S o , ( i v )     i s     m a t c h e d     w i t h     ( f ) . ( v ) A × ( B ∩ C ) = ( A × B ) ∩ ( A × C )                       S o , ( v )     i s     m a t c h e d     w i t h     ( d ) . ( v i ) A × ( B ∪ C ) = ( A × B ) ∪ ( A × C )                       S o , ( v i )     i s     m a t c h e d     w i t h     ( e ) .

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