The set $(A \cup B \cup C) \cap {(A \cap B^{'} \cap C^{'})}^{'} \cap C^{'}$ is equal to

A \cap B

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A \cap C^{'}

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B \cap C^{'}

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B^{'} \cap C^{'}

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Solution

The correct option is B $B \cap C^{'}$
$(A \cup B \cup C) \cap {(A \cap B^{'} \cap C^{'})}^{'} \cap C^{'}$
$= (A \cup B \cup C) \cap (A^{'} \cup B \cup C) \cap C^{'}$
$= {(A \cap A^{'}) \cup (B \cup C)} \cap C^{'}$
$= (ϕ \cup B \cup C) \cap C^{'} = (B \cup C) \cap C^{'}$
$= (B \cap C^{'}) \cup (C \cap C^{'})$
$= (B \cap C^{'}) \cup ϕ = B \cap C^{'}$

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