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If ๐‘ˆ={1,2,3,4,5,6,7,8,9,10,11}U={1,2,3,4,5,6,7,8,9,10,11}, ๐ด={1,3,5,8}A={1,3,5,8} and ๐ต={2,3,5,7,11}B={2,3,5,7,11}, then what is order of the power set ๐‘ƒ(๐‘€)P(M), where ๐‘€=๐ดโˆฉ๐ตโ€ฒM=AโˆฉB โ€ฒ ?

Question

If ๐‘ˆ={1,2,3,4,5,6,7,8,9,10,11}U={1,2,3,4,5,6,7,8,9,10,11}, ๐ด={1,3,5,8}A={1,3,5,8} and ๐ต={2,3,5,7,11}B={2,3,5,7,11}, then what is order of the power set ๐‘ƒ(๐‘€)P(M), where ๐‘€=๐ดโˆฉ๐ตโ€ฒM=AโˆฉB โ€ฒ ?

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Solution

To solve this problem, we first need to understand what each symbol represents:

  • ๐‘ˆ represents the universal set.
  • ๐ด and ๐ต are subsets of ๐‘ˆ.
  • ๐ตโ€ฒ represents the complement of ๐ต in ๐‘ˆ, which includes all elements in ๐‘ˆ that are not in ๐ต.
  • ๐‘€=๐ดโˆฉ๐ตโ€ฒ represents the intersection of sets ๐ด and ๐ตโ€ฒ, which includes all elements that are in both ๐ด and ๐ตโ€ฒ.
  • ๐‘ƒ(๐‘€) represents the power set of ๐‘€, which includes all possible subsets of ๐‘€. The order of a power set is 2^n, where n is the number of elements in the set.

Now, let's solve the problem step by step:

  1. Find ๐ตโ€ฒ: As ๐ต={2,3,5,7,11}, ๐ตโ€ฒ includes all elements in ๐‘ˆ that are not in ๐ต. So, ๐ตโ€ฒ={1,4,6,8,9,10}.

  2. Find ๐‘€=๐ดโˆฉ๐ตโ€ฒ: As ๐ด={1,3,5,8} and ๐ตโ€ฒ={1,4,6,8,9,10}, ๐‘€ includes all elements that are in both ๐ด and ๐ตโ€ฒ. So, ๐‘€={1,8}.

  3. Find the order of ๐‘ƒ(๐‘€): As ๐‘€={1,8}, it has 2 elements. So, the order of ๐‘ƒ(๐‘€) is 2^n = 2^2 = 4.

This problem has been solved

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