FORMULAS USED IN SET OPERATIONS

For any two finite sets A and B, we have the following useful results

(i) n(A) = n(A - B) + n(A n B)

(ii) n(B) = n(B - A) + n(A n B)

(iii) n(A U B) = n(A - B) + n(A n B) + n(B - A)

(iv) n(A U B) = n(A) + n(B) - n(A n B)

(v) n(A U B) = n(A) + n(B), when A n B = null set

(vi) n(A) + n(A') = n(U)

Example 1 :

If n(A n B) = 5, n(A U B) = 35, n (A) = 13, find n (B) .

Solution :

Using the formula,

n(A U B) = n(A) + n(B) - n(A n B)

35 = 13 + n (B) - 5

 35  =  8 + n (B) 

Subtract 8 on both sides

35 - 8 = 8 + n (B) - 8

27 = n (B) 

Hence the value of n (B) is 27.

Example 2 :

If n (A) = 26, n (B) = 10, n (A U B) = 30, n (A') = 17,   find n (A n B) and n (U) .

Solution :

Using the formula,

n(A U B) = n(A) + n(B) - n(A + B)

30 = 26 + 10 - n(A n B)

30 = 36 - n(A n B)

Subtract 36 on both sides

30 - 36 = 36 - n(A n B) - 36

-6 = - n(A n B)

Hence the value of n (A n B) is 6

n (U) = n (A) + n (A')

         =  26 + 17  ==> n (U) = 43

Hence the value of n (U) is 43.

Example 3 :

If n(U) = 38, n(A) = 16, n(A n B) = 12, n(B') = 20

find n(A U B).

Solution :

n(A U B)  =   n(A) + n(B) - n(A n B)

In order to use the above formula to find n (AUB), we need the value of n (B).

n (B) + n (B')  =  n (U)

n (B) + 20  =  38

n (B) = 38 - 20  ==>  18

n(A U B)  =   16 + 18 - 12

  =  34 - 12

  =  22

Hence the value of n(A U B) is 22.

Example 4 :

A and B are two sets such that n(A - B) = 32 + x, n(B - A) = 5x and n(A n B) = x Illustrate the information by means of a Venn diagram. Given that n(A) = n(B) . Calculate (i) the value of x (ii) n(A U B) .

Solution :

Formula to find n (A) is  n(A) = n(A - B) + n(A n B)

Formula to find n (B) is  n(B) = n(B - A) + n(A n B)

 n(A) = 32 + x + x ==>  32 + 2x  ---(1) 

 n(B) = 5x + x  ==> 6x -----(2)  

Given that n (A) = n (B)

 32 + 2x  =  6x

Subtract 2x on both sides

32 + 2x - 2x = 6x - 2x

 32  = 4x

Divide by 4 on both sides, we get

32/4  =  4x /4

x = 8

(i) Hence the value of x is 8.

(ii) n(A U B) = n(A - B) + n(A n B) + n(B - A)

  =  32 + x + x + 5x 

  =  32 + 7x

Applying the value of x,

  =  32 + 7(8)

  =  32 + 56

  =  88

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