Question

38. Consider three sets \( X, Y \) and \( Z \) having 6,5 and 4 elements respectively. A.11 these 15 elements are distinct. Let \( S=(X-Y) \cup Z \). How many proper subsets does \( S \) have? (a) 255 (b) 256 (c) 1023 (d) 1024

Answer 1

Given :

n(X)=6 

n(y)=5 

n(z)=4

    Also,the elements are distinct 

Therefore,these three are disjoint sets

==>n(X∩Z) =0 -------- (1)

Now,her

S=(X-Y)∪Z=X∪Z [Because X∩Z=∅==>X-Z=X]

==>n(S)=n(X∪Z)

==>n(S)=n(X)+n(Z)-n(X∩Z)

==>n(S)=n(X)+n(Z)-0 [From (1)]

==>n(S)=6+4=10

Therefore,

Number of proper subsets of S=2^n(S)-1

                                                                 =2¹⁰-1

                                                      =1024-1=1023

Hence,the correct answer is option (c)1023