Answer
$S$ spans $R^3$.
Work Step by Step
Assume the combination
$$a(1,0,1)+b(1,1,0)+c(0,1,1)=(0,0,0).$$
We have the system
\begin{align*}
a+b&=0\\
b+c&=0\\
a+c&=0
\end{align*}
Since the determinant of the matrix is given by
$$\left| \begin{array} {cc} 1&1&0\\0&1&1\\1&0&1 \end{array} \right|=2 \neq 0$$
then the system has unique solution, that is, $$a=0, \quad b=0, \quad c=0.$$
Consequently, $S$ is linearly independent and since $R^3$ has the dimension $3$ then $S$ spans $R^3$.