The number of minterms after minimizing the following Boolean expression $[D^{'} + A B^{'} + A^{'} C + A C^{'} D + A^{'} C^{'} D]^{'}$
is _____

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Solution

$[D^{'} + A B^{'} + A^{'} C + A C^{'} D + A^{'} C^{'} D]^{'}$

$= [D^{'} + A C^{'} D + A B^{'} + A^{'} C + A^{'} C^{'} D]^{'}$
$= [D^{'} + A C^{'} + A B^{'} + A^{'} [C + C^{'} D]]^{'}$
$= [D^{'} + A C^{'} + A B^{'} + A^{'} [C + D]]^{'}$
$= [D^{'} + A C^{'} + A B^{'} + A^{'} C + A^{'} D]^{'}$
$∵ (D^{'} + A^{'} D = D^{'} + A^{'})$
$= [D^{'} + A^{'} + A C^{'} + A B^{'} + A^{'} C]^{'}$
$∵ A^{'} + A^{'} C + = A^{'}$
$∵ A^{'} + A C^{'} + A B^{'} = A^{'} + A (C^{'} + B^{'}) = A^{'} + C^{'} + B^{'}$
$= [D^{'} + A^{'} + C^{'} + B^{'}]^{'}$
$= A B C D$
Hence, only $1$ minterm is required.

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