Question

Let A = {$x:xϵN$}, B = {$x:x=2n,nϵN$}, C = {$x:x=2n-1,nϵN$} and , D = {x: x is a prime natural number}. Find :

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

(iii) $A\cap D$ (iv) $B\cap C$

(v) $B\cap D$ (vi) $C\cap D$

Answer 1

(i) We have ,

A : {$x:xϵN$}

= {1, 2, 3,......}, the set of natural numbers

B= {$x:x=2n,xϵN$}

= {2,4,6,8,....}, the set of even natural numbers

$\therefore A\cap B$ = {$x:xϵAandxϵB$}

= {2,4,6,.....}

= B [$\therefore B\subset A$]

(ii) We have ,

A {$x:xϵN$}

= {1,2,3,.....}, the set of natural numbers

C = {$x:x=2n-1,xϵN$}

= {1,3,5,.....}, the set of odd natural numbers

$A\cap C$ = {$x:xϵAandxϵC$}

= C [$\therefore C\subset A$]

(iii) We have,

A { $x:xϵN$}

= {1,2,3,....}, the set of natural numbers

and D = {x : x is a prime natural number}

= {2,3,5,7}

$A\cap D$= {$x:xϵAandxϵD$}

= D [ $\therefore D\subset A$]

(iv) We have

B = {$x:x=2n,xepsilonN$}

= {2,4,6,8,.....}, the set of even natural numbers

and

C= {$x:x=2n-1,xϵN$}

= {1,3,5,....} , the set of odd natural numbers

$B\cap C$ = {$x:xϵBandxϵC$}

= $\varphi$ $\left[\begin{array}{c}\therefore \text{B and C are disjoint sets, i.e.,}\\ \text{have no elements in common}\end{array}\right]$

(v) Here, B = {$x:x=2n,xϵN$}

= {2,4,6,8,...}, the set of even natural numbers

and D = {x :x is a prime natural number}

= {2,3,5,7,....}

$B\cap D$ = {$x:xϵBandxϵD$}

= {2}

(vi) Here,

C = {$x:x=2n-1,xepsilonN$}

= {1,3,5,...}, the set of odd natural numbers

and D = {x :x is a prime natural number}

= {2,3,5,7,...}

$C\cap D$= {$x:xϵCandxϵD$}

We observe that except, the element 2, every other element in D is an odd natural number.

Hence, $C\cap D=$ D - {2}

= {$xϵD:x\ne 2$}