Question

(i) If A = {1,2,3,4,5},

B= {4,5,6,7,8},

C= {7,8,9,10,11} and

D= {10,11,12,13,14}. Find :

(i) $A\cup B$ (ii) $A\cup C$

(iii) $B\cup C$ (iv) $B\cup D$

(v) $A\cup B\cup C$ (vi) $A\cup B\cup D$

(vii) $B\cup C\cup D$ (viii) $A\cap (B\cup C)$

(ix) $(A\cap B)\cap (B\cap C)$

(x) $(A\cup D)\cap (B\cup C)$

Answer 1

(i) A = {1,2,3,4,5}

B= {4,5,6,7,8}

So, $A\cup B$ = {$x:xϵAorxϵB$}

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

(ii) $A\cup C$ = {$x:xϵAorxϵC$}

= {1,2,3,4,5,6,7,8,9,10,11}

(iii) $B\cup C$ = {$x:xϵBorxϵC$}

= {4,5,6,7,8,9,10,11}

(iv) $B\cup D$ = {$x:xϵBorxϵD$}

= {4,5,6,7,8,9,10,11,12,13,14}

(v) $A\cup B\cup C$ = {$x|xϵAorxϵBorxϵC$}

= {1,2,3,4,5,6,7,8,910,11}

(vi) $A\cup B\cup D$ = {$x|xϵBorxϵCorxϵD$}

= {4,5,6,7,8,9,10,11,12,13,14}

(viii) $A\cap (B\cup C)$ = all those elements which are common to A and $B\cup C$

= {$x|xϵAandxϵB\cup C$}

Now, $B\cup C$ = {4,5,6,7,8,9,10,11}

$\therefore A\cap (B\cup C)$ = {1,2,3,4,5} \cap {4,5,67,8,9,10,11}

= {4, 5}

(ix) $(A\cap B)\cap (B\cap C)$ = {$x|xϵ(A\cap B)andxϵ(B\cap C)$}

Now,

$A\cap B$ = {$x|xepsilonAandxϵB$}

i.e., elements which are common to A and B

$\therefore$ $A\cap B$ = {1,2,3,4,5} $\cap$ {4,5,6,7,8}

= {4, 5}

Also,

$B\cap C$= {4,5,6,7,8} $\cap$ {7,8,9,10,11}

= {7,8}

Hence, $(A\cap B)\cap (B\cap C)=4.5\cap 7,8$

= $\varphi$

$\left[\begin{array}{c}\therefore thereisnoelementcommonin\\ \{4,5\}and\{7,8\}\end{array}\right]$

(x) $(A\cup D)\cap (B\cup C)$ = {$x|xϵ(A\cup D)orxϵ(B\cup C)$}

Now,

$A\cup D$ = {1,2,3,4,5,10,11,12,13,14} and $B\cup C$ = {4,5,6,7,8,9,10,11}

$\therefore (A\cup D)\cap$ {$B\cup C$}= {4,5,10,11}