If ๐={1,2,3,4,5,6,7,8,9,10,11}U={1,2,3,4,5,6,7,8,9,10,11}, ๐ด={1,3,5,8}A={1,3,5,8} and ๐ต={2,3,5,7,11}B={2,3,5,7,11}, then what is order of the power set ๐(๐)P(M), where ๐=๐ดโฉ๐ตโฒM=AโฉB โฒ ?
Question
If ๐={1,2,3,4,5,6,7,8,9,10,11}U={1,2,3,4,5,6,7,8,9,10,11}, ๐ด={1,3,5,8}A={1,3,5,8} and ๐ต={2,3,5,7,11}B={2,3,5,7,11}, then what is order of the power set ๐(๐)P(M), where ๐=๐ดโฉ๐ตโฒM=AโฉB โฒ ?
Solution
To solve this problem, we first need to understand what each symbol represents:
- ๐ represents the universal set.
- ๐ด and ๐ต are subsets of ๐.
- ๐ตโฒ represents the complement of ๐ต in ๐, which includes all elements in ๐ that are not in ๐ต.
- ๐=๐ดโฉ๐ตโฒ represents the intersection of sets ๐ด and ๐ตโฒ, which includes all elements that are in both ๐ด and ๐ตโฒ.
- ๐(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:
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Find ๐ตโฒ: As ๐ต={2,3,5,7,11}, ๐ตโฒ includes all elements in ๐ that are not in ๐ต. So, ๐ตโฒ={1,4,6,8,9,10}.
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Find ๐=๐ดโฉ๐ตโฒ: As ๐ด={1,3,5,8} and ๐ตโฒ={1,4,6,8,9,10}, ๐ includes all elements that are in both ๐ด and ๐ตโฒ. So, ๐={1,8}.
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Find the order of ๐(๐): As ๐={1,8}, it has 2 elements. So, the order of ๐(๐) is 2^n = 2^2 = 4.
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