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
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
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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=2n(S)-1
=210-1
=1024-1=1023
Hence,the correct answer is option (c)1023
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