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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) $(A \cup B)^{\prime}=A^{\prime} \cap B^{\prime}$
(ii) $(A \cap B)^{\prime}=A^{\prime} \cup B^{\prime}$
(i) $(A \cup B)^{\prime}=A^{\prime} \cap B^{\prime}$
(ii) $(A \cap B)^{\prime}=A^{\prime} \cup B^{\prime}$
Solution:
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Verified Answer
(i) $A \cup B=\{2,4,6,8\} \cup\{2,3,5,7\}$ $=\{2,3,4,5,6,7,8\}$
L.H.S. $=(\mathrm{A} \cup \mathrm{B})^{\prime}=\mathrm{U}-(\mathrm{A} \cup \mathrm{B})$
$=\{1,2,3,4,5,6,7,8,9\}-\{2,3,4,5,6,7,8\}=\{1,9\}$
Now, $\mathrm{A}^{\prime}=U-A$
$=\{1,2,3,4,5,6,7,8,9\}-\{2,4,6,8\}$
$=\{1,3,5,7,9\}$
and $B^{\prime}=U-B$
$=\{1,2,3,4,5,6,7,8,9\}-\{2,3,5,7\}$
$=\{1,4,6,8,9\}$
R.H.S. $=A^{\prime} \cap B^{\prime}$
$=\{1,3,5,7,9\} \cap\{1,4,6,8,9\}=\{1,9\}$
L.H.S. = R.H.S,
Hence $(A \cup B)^{\prime}=A^{\prime} \cap B^{\prime}$ is verified.
(ii) $A \cap B=\{2,4,6,8\} \cap\{2,3,5,7\}=\{2\}$
L.H.S. $=(A \cap B)^{\prime}=U-(A \cap B)$
$=\{1,2,3,4,5,6,7,8,9\}-\{2\}$
$=\{1,3,4,5,6,7,8,9\}$
R.H.S. $=A^{\prime} \cup B^{\prime}$
$=\{1,3,5,7,9\} \cup\{1,4,6,8,9\}$
$=\{1,3,4,5,6,7,8,9\}$
L.H.S = R.H.S.
Hence, $(A \cap B)=A^{\prime} \cup B^{\prime}$ is verified.
L.H.S. $=(\mathrm{A} \cup \mathrm{B})^{\prime}=\mathrm{U}-(\mathrm{A} \cup \mathrm{B})$
$=\{1,2,3,4,5,6,7,8,9\}-\{2,3,4,5,6,7,8\}=\{1,9\}$
Now, $\mathrm{A}^{\prime}=U-A$
$=\{1,2,3,4,5,6,7,8,9\}-\{2,4,6,8\}$
$=\{1,3,5,7,9\}$
and $B^{\prime}=U-B$
$=\{1,2,3,4,5,6,7,8,9\}-\{2,3,5,7\}$
$=\{1,4,6,8,9\}$
R.H.S. $=A^{\prime} \cap B^{\prime}$
$=\{1,3,5,7,9\} \cap\{1,4,6,8,9\}=\{1,9\}$
L.H.S. = R.H.S,
Hence $(A \cup B)^{\prime}=A^{\prime} \cap B^{\prime}$ is verified.
(ii) $A \cap B=\{2,4,6,8\} \cap\{2,3,5,7\}=\{2\}$
L.H.S. $=(A \cap B)^{\prime}=U-(A \cap B)$
$=\{1,2,3,4,5,6,7,8,9\}-\{2\}$
$=\{1,3,4,5,6,7,8,9\}$
R.H.S. $=A^{\prime} \cup B^{\prime}$
$=\{1,3,5,7,9\} \cup\{1,4,6,8,9\}$
$=\{1,3,4,5,6,7,8,9\}$
L.H.S = R.H.S.
Hence, $(A \cap B)=A^{\prime} \cup B^{\prime}$ is verified.
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