Question

If $X=\{8^n-7n-1:n\in N\}$ and $Y=\{49n-49:n\in N\}$. Then,

• A. $X\cap Y=\varphi$
• B. $X\subset Y$
• C. X = Y
• D. $Y\supset X$

Answer 1

The correct option is B $X\subset Y$

Given,
$X=\{8^n-7n-1:n\in N\}$ and $Y=\{49n-49:n\in N\}$

Putting $n=1$
$X=8-7-1=0$

Putting $n=2$
$X=8^2-7\times 2-1$

$X=64-15=49$

Putting $n=3$

$X=8^3-7\times 3-1$

$X=512-22=490$

So,
$X=\{0,49,\mathrm{490...}\}....\left(1\right)$

Now, $Y=\{49n-49:n\in N\}$

Putting $n=1$

$Y=49\times 1-49=0$

Putting $n=2$

$Y=49\times 2-49=49$

Putting $n=3$

$Y=49\times 3-49$

$=147-49=98$
.
.
.

Putting $n=11$

$Y=49\times 11-49$
$=539-49=490$

So,
$Y=\{0,49,\mathrm{98...490...}\}....\left(2\right)$

From $\left(1\right)\&\left(2\right)$

Every element of $X$ is in $Y.$
Therefore,

$X\subset Y$.

Thus, the correct answer is $\left(A\right).$