If U = {2, 3, 5, 7, 9} is the universal set
Question:

If U = {2, 3, 5, 7, 9} is the universal set and A = {3, 7}, B = {2, 5, 7, 9}, then prove that:

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

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

Solution:

Given:

U = {2, 3, 5, 7, 9}

A = {3, 7}

B = {2, 5, 7, 9}

To prove :

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

(ii) $(A \cap B)^{\prime}=A^{\prime} \cup B$,

Proof :

(i) LHS:

$(A \cup B)=\{2,3,5,7,9\}$

$(A \cup B)^{\prime}=\phi$

RHS

$A^{\prime}=\{2,5,9\}$

$B^{\prime}=\{3\}$

$A^{\prime} \cap B^{\prime}=\phi$

LHS = RHS

$\therefore(A \cup B)^{\prime}=A^{\prime} \cap B^{\prime}$

(ii) LHS

$(A \cap B)=\{7\}$

 

$(A \cap B)^{\prime}=\{2,3,5,9\}$

RHS

$A^{\prime}=\{2,5,9\}$

$B^{\prime}=\{3\}$

$A^{\prime} \cup B^{\prime}=\{2,3,5,9\}$

LHS = RHS

$\therefore(A \cap B)^{\prime}=A^{\prime} \cup B^{\prime}$

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