45. \( \{(x, y) \mid x y<0\} \) 47. \( \{(x, y)||x| \leqslant 2\} \)

The first set, \( \{(x, y) \mid x y < 0\} \), represents points in the Cartesian plane where the values of \(x\) and \(y\) have opposite signs. This means the points fall in the first and third quadrants (where both are positive) and the second and fourth quadrants (where both are negative). You can visualize it as a "V" shaped area that does not include the x-axis or y-axis. The second set, \( \{(x, y) \mid |x| \leqslant 2\} \), describes a vertical strip on the Cartesian plane where the x-values lie between -2 and 2 inclusively. Thus, all points that fall within this range will be represented, regardless of their y-values, creating an area that stretches infinitely up and down between those vertical lines at \(x = -2\) and \(x = 2\). This area helps encapsulate various functions that may vary in y while being confined in x.