Question

U = {0, 1, 2, 3, 4, 5, 6, 7}, A = {1, 3, 5, 7} மற்றும் B = { 0, 2, 3, 5, 7} எனில்‌ பின்வரும்‌ கணங்களைக்‌ காண்க.
(i) A’
(ii) B’
(iii) A'∪B'
(iv) A'∩B'
(v) (A∪B)'
(vi) (A∩B)'
(vii) (A’)’
(viii) (B’)’

Answer 1

U = {0, 1, 2, 3, 4, 5, 6, 7}, A = {1, 3, 5, 7} மற்றும் B = { 0, 2, 3, 5, 7}:  

(i) A'

$\begin{aligned}&\mathrm{U}-\mathrm{A}=\{0,1,2,3,4,5,6,7\}-\{1,3,5,7\}\\&=\{0,2,4,6\}\end{aligned}$

(ii) B'  

$\begin{aligned}&\mathrm{U}-\mathrm{B}=\{0,1,2,3,4,5,6,7\}-\{0,2,3,5,7\}\\&=\{1,4,6\}\end{aligned}$

(iii) A'∪B'    

$\begin{aligned}&A^{\prime} U B^{\prime}=\{0,2,4,6\} \cup\{1,4,6\}\\&=\{0,1,2,4,6\}\end{aligned}$

(iv) A'∩B'

$\begin{aligned}&A^{\prime} \cap B^{\prime}=\{0,2,4,6\} \cap\{1,4,6\}\\&=\{4,6\}\end{aligned}$

(v) (A∪B)'

$(A \cup B)^{\prime}=\{1,-3,5,7\} \\ {\{0,2,3,5,7\} =\{0,1,2,3,5,7\}\\ (A \cup B)^{\prime}=U-(A B) \\ =\{0,1,3,4,5,6,7\}- \{0,1,2,3,5,7\}\\ =\{4,6\}$

(vi) (A∩B)'

$\begin{aligned}&A \cap B=\{1,3,5,7\} \cap\{0,2,3,5,7\}\\&=\{3,5,7\}\end{aligned}$

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

$\begin{aligned}&=\{0,1,2,3,4,5,6,7\}-\{3,5,7\}\\&=\{0,1,2,4,6\}\end{aligned}$

(vii) (A')'

$\begin{aligned}&\left(A^{\prime}\right)^{\prime}-U-A^{\prime}=\{0,1,2,3,4,5,6,7\}-\{0,2,4,6\}\\&=\{1,3,5,7\}=\mathbf{A}\end{aligned}$

(viii) (B')'

$\begin{aligned}&\left(\mathrm{B}^{\prime}\right)^{\prime}=\mathrm{U}-\mathrm{B}^{\prime}=\{0,1,2,3,4,5,6,7\}-\{1,4,6\}\\&=\{0,2,3,5,7\}=B\end{aligned}$