Question

If X = {$4^n-3n-1:ϵN$} and
Y = {$9(n-1):ϵN$}
then $X\subset Y$

• A. True
• B. False

Answer 1

The correct option is A True

Given $X=\{4^n-3n-1;n\in N\}$

For $n=0$, $X=4^0-0-1$

$\therefore X=0$

For $n=1$, $X=4^1-3\left(1\right)-1$

$\therefore X=0$

For $n=2$, $X=4^2-3\left(2\right)-1$

$\therefore X=16-6-1=9$

For $n=3$, $X=4^3-3\left(3\right)-1$

$\therefore X=64-9-1=54$

For $n=4$, $X=4^4-3\left(4\right)-1$

$\therefore X=256-12-1=243$

$\therefore X=\{0,0,9,54,\mathrm{243.......}\}$ (1)

Consider second set $Y=\{9(n-1);n\in N\}$

For $n=0$, $Y=9(0-1)$

$\therefore Y=-9$

For $n=1$, $Y=9(1-1)$

$\therefore Y=0$

For $n=2$, $Y=9(2-1)$

$\therefore Y=9$

For $n=3$, $Y=9(3-1)$

$\therefore Y=18$

For $n=4$, $Y=9(4-1)$

$\therefore Y=27$

$\therefore Y=\{-9,0,9,18,27,36,45,54,63,\mathrm{72.........}\}$

$\therefore Y=\{y:yismultipleof9andy\ge -9\}$ (2)

From (1) and (2),

$X\subset Y$