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

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

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

 

Solution:

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

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

(i)

$(\mathrm{A} \cup \mathrm{B})^{\prime}=\{2,3,4,5,6,7,8\}^{\prime}=\{1,9\}$

$\mathrm{A}^{\prime} \cap \mathrm{B}^{\prime}=\{1,3,5,7,9\} \cap(1,4,6,8,9)=\{1,9\}$

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

(ii)

$(A \cap B)^{\prime}=\{2\}^{\prime}=\{1,3,4,5,6,7,8,9\}$

$\mathrm{A}^{\prime} \cup \mathrm{B}^{\prime}=\{1,3,5,7,9\} \cup\{1,4,6,8,9\}=\{1,3,4,5,6,7,8,9\}$

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

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