12.0 Number of elements in sets

  Set Theory

    12.0 Number of elements in sets

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12.0 Number of elements in sets$$\begin{equation} \begin{aligned} i.{\mkern 1mu} \;n(A \cup B) = n(A) + n(B) - n(A \cap B) \\ ii.\;\;n(A - B) = n(A) - n(A \cap B) = n(A \cup B) - n(B) \\ iii.\;\;n(A{\mkern 1mu} \Delta \;B) = n(A - B) \cup \;n(B - A) = n(A) - n(A \cap B) + n(B) - n(A \cap B) \\ \Rightarrow n(A{\mkern 1mu} \Delta \;B) = n(A) + n(B) - 2n(A \cap B) = n(A \cup B) - n(A \cap B) \\ iv.\;\;n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(A \cap C) + n(A \cap B \cap C) \\ v.\;\;Number\;of{\mkern 1mu} elements\;in\;exactly\;two\,{\mkern 1mu} of\;the\;sets \\ = n(A \cap B) + n(B \cap C) + n(A \cap C) - 3n(A \cap B \cap C) \\ vi.{\mkern 1mu} \;Number\;of{\mkern 1mu} elements\;in\;exactly\;one\;of\;the{\mkern 1mu} \;sets \\ = n(A) + n(B) + n(C) - 2n(A \cap B) - 2n(B \cap C) - 2n(A \cap C) + 3n(A \cap B \cap C) \\\end{aligned} \end{equation} $$