Question

Decide if the following statements are true or false. Explain. $$ \left\\{(x, y) \in \mathbb{R}^{2}: x-1=0\right\\} \subseteq\left\\{(x, y) \in \mathbb{R}^{2}: x^{2}-x=0\right\\} $$

Short answer

The statement is true. Every point in the set \{(x, y) \in \mathbb{R}^{2}: x - 1 = 0\} is also in the set \{(x, y) \in \mathbb{R}^{2}: x^{2} - x = 0\}, which makes the first set a subset of the second.

Step-by-step solution

Solve the First Equation

Analyze and solve the equation \(x - 1 = 0\). This equation simplifies to \(x = 1\). This means that all points with an x-coordinate of 1 lie in the first set.

Solve the Second Equation

Analyze and solve the equation \(x^{2} - x = 0\). This equation simplifies to \(x(x - 1) = 0\). This has two solutions, \(x = 0\) and \(x = 1\). This means that all the points with an x-coordinate of 0 or 1 lie in the second set.

Comparison of the Sets

The points in the first set, \((1, y)\), where \(y \in \mathbb{R}\), also exist in the second set since \(x = 1\) is a solution for the second equation. Thus, every point that is in the first set is also in the second set.