If S is the set of distinct values of b for which the following...

For no solution D = \left|\begin{cases}1 & 1 & 1 \\ 1 & a & 1 \\ a & b & 1\end{cases}\right| = 0

Expanding the determinant along the first row, we get $D = - {\left(a - 1\right)}^{2} = 0$

$⇒ a = 1$

For $a = 1$, first two planes are coincident.

Now, we have $x + y + z = 1 \text{ and } ax + by + z = 0$

For no solution, these two planes are parallel.

$⇒ \dfrac{1}{a} = \dfrac{1}{b} = \dfrac{1}{1}$

$⇒ a = 1 \text{ and } b = 1$.

$∴ b = \left\{1\right\} ⇒$it is a singleton set