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Application of Sets

Step 1
If A is a finite set, then the number of elements in A is denoted by n(A). If A and B are two finite disjoint sets, i.e. A  B = φ, then n(A  B) = n(A) + n(B).

The two sets that are shaded are A  B' and A  B. (Recall that A  B' consists of all the elements of A that do not lie in B.)

What is the intersection of A
 B' and A  B?

It is φ ? Thus, we can say that A  B' and A  B are two disjoint sets. 
Next, let us find the union of A  B' and A  B. From the figure it is clear that the union of these two sets is A. Thus, n(A) = n(A  B) + n(A  B). 
⇒ n (A  B') = n (A) - n (A  B) 

Step 2
Look at figure below:
The two sets that are shaded are A  B' and B. It is clear from the figure that A  B' and B are two disjoint sets and their union equals A  B.
Thus,
n(A  B) = n(A  B') + n(B) = n(A) - n(A  B) + n(B)     [by step 1] 
 n(A  B) = n(A) + n(B) - n(A  B) 
It immediately follows that
n(A  B) = n(A) + n(B) - n(A  B)

 

Examples
(i) If X and Y are two sets such that n(X) = 17, n(Y) = 23, n(X  Y) = 38, find n(X  Y).

Solution
We are given n(X) = 17, n(Y) = 23 and n (X  Y) = 38. Using the formula
n(X  Y) = n(X) + n(Y) - n(X  Y) we get
n(X  Y) = 17 + 23 - 38 = 2.

(ii) If X and Y are two sets such that X  Y has 18 elements, X has 8 elements, and Y has 15 elements, how many elements does X  Y have?  

Solution
(ii) We are given n(X  Y) = 18, n(X) = 8, n(Y) = 15. Using the formula
n(X  Y) = n(X) + n(Y) - n(X  Y) we get
n(X  Y) = 8 + 15 - 18 = 5.

(iii) If A and B are two sets such that A has 40 elements, A  B has 60 elements and A  B has 10 elements, how many elements does B have? 

Solution
(iii) We are given n(A) = 40, n(A  B) = 60 and n(A  B) = 10. Putting these values in the formula
n(A  B) = n(A) + n(B) - n(A  B) we get
60 = 40 + n(B) - 10 ⇒ n(B) = 30. 

(iv) In a group of 70 people, 37 like coffee, 52 like tea and each person likes either coffee or tea. Find how many like (a) both coffee and tea. (b) tea but not coffee? 

Solution
(iv) Let C, T denote the set of persons who like coffee, tea respectively.
We are given n(C) = 37 and n(T) = 52. Since each of the 70 people like either coffee or tea, 
n(C  T) = 70. Using the formula
n(C  T) = n(C) + n(T) - n(C  T)
We get 70 = 37 + 52 - n(C  T) ⇒ n(C  T) = 19. 
The people who like tea but not coffee lie in the set C'  T.  We have
n(C'  T) = n(T) - n(C  T)
              = 52 - 19 = 33
(v) In a group of 50 people, 35 speak Hindi, 25 speak both English and Hindi, and all the people speak at least one of two languages. Find the number of people who speak English but not Hindi.

Solution
(v) Let H denote the set of people who speak Hindi and E denote the set of people who speak English. We are given n(H) = 35 and n(E ∩ H) = 25. Since each of the person from the group either speaks Hindi or English, we have n(H  E) = 50. We are interested to find the number of people who speak English but not Hindi, i.e. we wish to find n(E  H').

We know that
n(E  H') = n(E) - n(E  H)
But n(E  H) = n(E) + n(H) - n(E  H)
 n(E) - n(E  H) = n(E  H) - n(H)
= 50 - 35 = 15.
Thus, n(E  H') = 15.




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