If U = {1, 2, 3, 4, 5,6,7,8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}. Verify that (i) (ii)

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Solution

(i)

Given sets are,

U={ 1,2,3,4,5,6,7,8,9 } A={ 2,4,6,8 } B={ 2,3,5,7 } .

A∪B={ 2,3,4,5,6,7,8 } .

Therefore, ( A∪B ) ′ ={ 1,9 } .

A'={ 1,3,5,7,9 } .

B ′ ={ 1,4,6,8,9 } .

Therefore,

A ′ ∩ B ′ ={ 1,3,5,7,9 }∩{ 1,4,6,8,9 } ={ 1,9 } .

Hence, ( A∪B ) ′ = A ′ ∩ B ′ is verified.

(ii)

A∩B={ 2,4,6,8 }∩{ 2,3,5,7 } ={ 2 } .

Therefore,

( A∩B ) ′ =U−( A∩B ) ={ 1,2,3,4,5,6,7,8,9 }−{ 2 } ={ 1,3,4,5,6,7,8,9 } .

Now,

A ′ ={ 1,3,5,7,9 } and B ′ ={ 1,4,6,8,9 } .

Therefore, A ′ ∪ B ′ ={ 1,3,4,5,6,7,8,9 } .

Hence, ( A∩B ) ′ = A ′ ∪ B ′ is verified.

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Similar questions

U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8}

B = {2, 3, 5, 7}

(A \cup B)^{'} = A^{'} \cap B^{'}

If U = {1, 2, 3, 4, 5,6,7,8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}. Verify that

(i) (ii)

(A \cup B)' = A' \cap B'

(A \cap B)' = A' \cup B'

Let U = {1,2,3,4,5,6,7,8,9},

A = {2,4,6,8} and B = {2,3,5,7}.

Verify that :

(i) $(A \cup B)^{'} = A^{'} \cap B^{'}$

(ii) $(A \cap B)^{'} = A^{'} \cup B^{'}$ .

U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8}

B = {2, 3, 5, 7}

(A \cap B)^{'} = A^{'} \cup B^{'}

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