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Set theory formulas





General Sets Formula's

$A \cup A = A$
$A \cap A = A$
$A \cap A^c = \phi$
$A \cup A^c = U$
$A \cup \phi = A$
$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)
$=n(A \cap B) + n(B \cap C) + n(C \cap A) - 2 \times n( A \times B \times C)$



Quiz Time

Question 1 n(A)=10, n(B) =5 , $n(A \cap B) =0$, find the value of n(A -B)
A) 5
B) 15
C) 10
D) none of these
Question 2 which of these is false ? A) $A \cup A=A$
B) $(A \cup B)^c = A^c \cup B^c$
C) $A \cap B = B \cap A$
D) None of the above
Question 3 which of these is a empty set
A) $A \cup A$
B) $A \cap A$
C) $A \cup A^c$
D) $A \cap A^c$
Question 4 A = { 1,2,3,4,5} and B ={2,3,4,5,6,7} then A - B
A) {1,6,7}
B) {1}
C) {2,3,4,5,6,7}
D) {1,2,3,4,5,6,7}
Question 5 n(A) = 26 ,n(B) =20, $n(A \cup B) =40$ then $A \cap B$
A) 14
B) 20
C) 6
D)34
Question 6A ={ 2,3,4} and B ={1,3,7,2},then n(B -A) is
A. 1
B. 2
C. 4
D. 3


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