Question

From the given diagram, find :

$\left(i\right)A^{\prime }\left(ii\right)B^{\prime }\left(iii\right)A^{\prime }\cup B^{\prime }\left(iv\right)(A\cap B{)}^{\prime }$

$\text{Is}{A}^{\prime }\cup B^{\prime }=(A\cup B{)}^{\prime }?\text{Also, verify if}{A}^{\prime }\cap B^{\prime }=(A\cup B{)}^{\prime }.$

Answer 1

$\left(i\right)A=\{1,3,4,6\}\therefore A^{\prime }=\{2,5,7,8,9,10\}Ans.\left(ii\right)B=\{1,2,5\}\therefore B^{\prime }=\{3,4,6,7,8,9,10\}Ans.\left(iii\right)A^{\prime }\cup B^{\prime }=\{2,5,7,8,9,10\}\cup \{3,4,6,7,8,9,10\}=\{2,3,4,5,6,7,8,9,10\}\left(iv\right)A\cap B=\{1,3,4,6\}\cap \{1,2,5\}=\left\{1\right\}\therefore (A\cap B{)}^{\prime }=\{2,3,4,5,6,7,8,9,10\}Ans.\text{From part (iii) and part (iv) we conclude}{A}^{\prime }\cup B^{\prime }=(A\cap B{)}^{\prime }\text{Now}A\cap B=\{2,5,7,8,9,10\}\cup \{3,4,6,7,8,9,10\}\Rightarrow A^{\prime }\cap B^{\prime }=\{7,8,9,10\}......\left(I\right)\text{Now}A\cup B\}=\{1,3,4,6\}\cup \{1,2,5\}=\{1,2,3,4,5,6\}\therefore (A\cup B{)}^{\prime }=\{7,8,9,10\}.......\left(II\right)$

From I and II we conclude

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