Fsc Part 1 Mathematics (Complete Solution)

Q3
Verify De Morgan's Laws for the following sets

U={1,2,3,.....,20}, A={2,4,6,....,20} and {1,3,5,....,19}

SOLUTION:

(AUB)c=Ac∩Bc

first we find AUB

AUB={2,4,6,...,20}U{1,5,7,...,19}={1,2,3,4,5,6,7,....20}

then we find (AUB)c

(AUB)c=U-(AUB)

={1,2,3,4,..........20}-{1,2,3,4,.........,20}

={   }

now we find

Ac=U-A={1,2,3,4,....,20}-{2,4,6,......,20}={1,3,5,7,...19}

Bc=U-B={1,2,3,4,....,20}-{1,3,5,......19}={2,4,6,8,....20}

then we can find Ac∩Bc

Ac∩Bc={1,3,5,....,19}∩{2,4,6,...,20}={  }

hence (AUB)c=Ac∩Bc

Similarly               (A∩B)c=AcUBc

A∩B={2,4,6.....20}∩{1,5,7,....19}={  }

(A∩B)c=U-(A∩B)={1,2,3,......20}-{  }={1,2,3,.......20}

Now,

Ac={1,3,5,.......,19} ,Bc={2,4,6,......,20}

=> AcUBc={1,3,5,....,19}U{2,4,6,......20}={1,2,3,...20}

Hence (A∩B)c=AcUBc

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