$A \cup A = A$
$A \cap A = A$
$A \cap A^c = \phi$
$A \cup A^c = U$
$A \cup \phi = A$
$A \cap \phi = \phi $
$(A^c)^c= A$
$ A \cup B = B \cup A$
$A \cap B = B \cap A$
$A \cup (B \cup C) = ( A \cup B ) \cup C$
$A \cup ( B \cap C) = (A \cup B) \cap (A \cup C)$
$ (A \cup B) ^c = A^c \cap B^c$
$ (A \cap B) ^c = A^c \cup B^c$
$ A -(B \cup C)= (A -B) \cap (A-C)$
For two disjoint sets A and B
$n(A \cup B) = n(A) + n(B)$
$n(A - B) =n(A)$
$n( A \cap B) =0$
$n(B - A) =n(B)$
$n(U) = n(A) + n(B) + n( (A \cup B)^c )$
$ n(A) = n(A \cup B) - n(B)$
$ n(B) = n(A \cup B) - n(A)$
$n (A \Delta B) = n(A) + n(B)$
For twooverlappingsets A and B
$n(A \cup B) = n(A) + n(B) - n(A \cap B)$
$n(A - B) = n(A \cup B) - n(B)$
$n(A - B) = n(A ) - n(A \cap B)$
$n(B - A) =n(A \cup B) - n(A)$
$n(U) = n(A) + n(B) - n(A \cap B) + n( (A \cup B)^c )$
$ n(A) = n(A \cup B)+ n(A \cap B) - n(B)$
$ n(B) = n(A \cup B) + n(A \cap B) - n(A)$
$n (A\cup B) = n(A -B) + n(B -A) + n(A \cap B)$
$ n(A^c) = n(U) - n(A)$
For three overlapping sets A,B and C
$n(A \cup B \cup C)= n(A) + n(B) + n(C) – n(A \cap B) – n(A \cap C) – n(B \cap C) + n(A \cap B \cap C)$
$ n(A \cap B only) = n( A \cap B) - n(A \cap B \cap C$
$ n(A \cap C only) = n( A \cap C) - n(A \cap B \cap C$
$ n(B \cap C only) = n(B \cap C) - n(A \cap B \cap C$
$n(A only) = n(A) - n(A \cap B) - n(A \cap C) + n(A \cap B \cap C)$
$n(B only) = n(B) - n(A \cap B) - n(B \cap C) + n(A \cap B \cap C)$
$n(C only) = n(C) - n(B \cap C) - n(A \cap C) + n(A \cap B \cap C)$
$n(U) = n(A) + n(B) + n(C) – n(A \cap B) – n(A \cap C) – n(B \cap C) + n(A \cap B \cap C) + n( (A \cup B \cup C)^c )$ So, No of elements in exactly two of the sets
$=n(A \cap B) + n(A \cap C) + n(B \cap C) – 3 n(A \cap B \cap C) So,No of persons in exactly one set
$=n(A) + n(B) + n(C) – 2 \times n(A \cap B) – 2 \times n(A \cap C) – 2 \times n(B \cap C) + 3 \times n (A \cap B \cap C)$ So,No ofelements in two or more sets (at least 2 sets)
$=n(A \cap B) + n(B \cap C) + n(C \cap A) - 2 \times n( A \times B \times C)$
