1. State the problem: We are given the set $S = \{(x,y) \mid x,y \in \mathbb{R} \text{ and } y = 2x - 2\}$, which represents points $(x,y)$ in the real plane satisfying the linear equation $y = 2x - 2$.\n\n2. Understand the equation: The equation $y = 2x - 2$ describes a straight line where the slope (rate of change) is $2$ and the $y$-intercept (where the line crosses the $y$-axis) is $-2$. This means for every unit increase in $x$, $y$ increases by $2$.\n\n3. Find the $y$-intercept: When $x=0$, substituting into the equation gives $y = 2 \cdot 0 - 2 = -2$. So the line crosses the $y$-axis at $(0,-2)$.\n\n4. Find the $x$-intercept: When $y=0$, set $0 = 2x - 2$ and solve for $x$:\n$$0 = 2x - 2$$\n$$2x = 2$$\n$$x = 1$$\nSo the $x$-intercept is at $(1,0)$.\n\n5. Summary: The set $S$ represents all points on the line passing through $(0,-2)$ and $(1,0)$ with slope $2$.