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Question

If X = { 8n7n1:n ϵ N } and Y = {49(n1):n ϵ N}, then prove that XY.


Solution

X={8n7n1:x ϵ N} 

Y={4n(n1):n ϵ N}

In order to show that xY we show that every element of X is an element of Y.

So let x ϵ Xx=8m7m1 for some m ϵ N.

x=(1+7)m7m1
= { mC01m+mC11m17+...+mCm1117m1+mCm7m)7m1

  [using binomial expansion]=1+7m+mC272+mC373+....mCm7m7m1

=mC272+mC373+....mCm7m 

=49(mC2+mmC3+....+mCm7m2),m2

=49tm,m2, where tm=mC2+mC37+....+mCm7m2

Is some positive interger depending on m2 

For m = 1

x=817×11

= 8-8

= 0

Hence, X contains all positive integral multiples of 49.

Also, Y consists of all positive integral multiples of 49, including 0, for n = 1.

Thus, we conclude that XY


Mathematics
RD Sharma
Standard XI

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