Question

If $X=\{1,2,3,4,5,6,7,8,9,10\}$ is the universal set & $A=\{1,2,3,4\},B=\{2,4,6,8\},C=\{3,4,5,6\}$ verify the following.
a) $A\cup (B\cup C)=(A\cup B)\cup C$
b) $A\cap (B\cup C)=(A\cap B)\cup (A\cap C)$
c) ${\left(A^{\prime }\right)}^{\prime }=A$

Answer 1

$X=\{1,2,3,4,5,6,7,8,9,10\}$

$A=\{1,2,3,4\}$

$a$

$(B\cup C)=\{2,3,4,5,6,8\}$

$A\cup (B\cup C)=\{1,2,3,4,5,6,8\}----\left(1\right)$

$(A\cup B)=\left\{1.2.3.4.6.8\right\}$

$(A\cup B)\cup C=\{1,2,3,4,5,6,8\}----\left(2\right)$

$\therefore$ from $\left(1\right)$ and $\left(2\right)$

$A\cup (B\cup C)=(A\cup B)\cup C$

$b$

$(B\cup C)=\{2,3,4,5,6,8\}$

$A\cap (B\cup C)=\{2,3,4\}---\left(3\right)$

$(A\cap B)=\{2,4\}$

$(A\cap C)=\{3,4\}$

$\therefore (A\cap B)\cup (A\cap C)=\{2,3,4\}---\left(4\right)$

$\therefore$ from $\left(3\right)$ and $\left(4\right)$

$A\cap (B\cup C)=(A\cap B)\cup (A\cap C)----proved$

$c$

$A=\{5,6,7,8,9,10\}$

$(A^{\prime }{)}^{\prime }=\{1,2,3,4,\}=A---Proved$