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  4. The set (A ∪ B ∪ C) ∩ (A ∩ Bc ∩ Cc)c ∩ : : Cc is equal to

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Q. The set $(A \cup B \cup C) \cap (A \cap B^c \cap C^c)^c$ $\cap \: \: C^c$ is equal to

Sets

Solution:

$ (A \cup B \cup C) \cap (A \cup B^c \cup c^c)^c \cap C^c $
$ = (A \cup B \cup C)\cap(A^c\cup B\cup C)\cap C^c $
$= (A \cap A^c) \cup (B \cup C) \cap C^c$
$=\phi \cup(B \cup C) \cap C^c = (B \cup C) \cap C^c$
$=(B \cap C^c) \cup (C \cap C^c) = B \cap C^c \cup \phi$
$=B \cap C^c $