If the region of convergence of x1[n] + x2[n] is $${1 \over 3} < \left| z \right| < {2 \over 3},$$ then the region of convergence of x1[n] - x2[n] includes

”$${1
+ x_2[n]$ is $${1 \over 3} < \left| z \right| < {2 \over 3},$$ so the region of convergence of $x_1[n] – x_2[n]$ must be a subset of this region. Since the region of convergence of a sequence is always a disk centered at the origin, the only possible region of convergence for $x_1[n] – x_2[n]$ is $${1 \over 3} < \left| z \right| < {2 \over 3}.$$

Option A is incorrect because it includes the point $z=3$, which is not in the region of convergence of $x_1[n] – x_2[n]$. Option B is incorrect because it includes the point $z=2$, which is not in the region of convergence of $x_1[n] – x_2[n]$. Option C is incorrect because it includes the point $z=3/2$, which is not in the region of convergence of $x_1[n] – x_2[n]$.

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