how many subsets of 1,2,3,4,5,6,7,8,9 contain 5 consecutive numbers
Answers
Answer:
Total subsets = 2^8
Out of all the nos , 4 are prime and 4 are non prime.
Subsets without any prime = 2^4
Subsets with at least one prime = 256-16 = 240
Step-by-step explanation:
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Answer 1. Let S⊆X such that S∩A≠∅ . This is equivalent to saying that S is not a subset of B . In other words,
S∩A≠∅⇔S⊈B .
Therefore, the number of such sets S plus the number of subsets of B together equal the number of subsets of X . This number is 2n−2n−m . ■
Answer 2. Let S be as before. Then
S=S∩X=(S∩A)∪(S∩B)
is a disjoint union of S1=S∩A and S2=S∩B . So there are as many sets S as there are ordered pairs (S1,S2) of subsets of (A,B) . Set S1 is non-empty but there is no restriction on the set S2 . Therefore the number of such sets S is (2m−1)2n−m . ■
For the given problem, X={2,3,4,…,9} and A={2,3,5,7} . So n=8 , m=n−m=4 and the desired number of subsets is 28–24=24(24−1)=240 . ■