If $X=\left\{8^n-7n-1:n\in \mathbf{N}\right\}$ and $\mathrm{Y}=\left\{49\right(\mathrm{n}-1):\mathrm{n}\in \mathbf{N}\}$ then

$8^n-7n-1=(1+7{)}^n-7n-1$

Which is a multiple of 49 of $\mathrm{n}\ge 2$

For $n=1,8^n-7n-1=8-7-1=0$

For $\mathrm{n}=2,8^{\mathrm{n}}-7\mathrm{n}-1=64-14-1=49$

a multiple of 49. So, $8^n-7n-1$ is a multiple of 49 for all $\mathrm{n}\in \mathbf{N}$

Hence X contains all elements which are multiples of 49 for $\mathrm{n}\in \mathbf{N}$. Also Y contains all elements which are multiples of 49.

$\therefore \mathrm{X}\subseteq \mathrm{Y}$

Note that Y contains all multiples of 49 but X does not contain all multiples of 49 but some, i.e. $\mathrm{Y}⊈\mathrm{X}$