Chapter 1: Problem 6
Write out the indicated sets by listing their elements between braces. $$ \left\\{x \in \mathbb{R}: x^{2}=x\right\\} \times\left\\{x \in \mathbb{N}: x^{2}=x\right\\} $$
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Sketch the following sets of points in the \(x-y\) plane. $$ \left\\{(x, y): x, y \in \mathbb{R}, x^{2}+y^{2}=1\right\\} $$
Decide if the following statements are true or false. Explain. $$ \mathbb{R}^{2} \subseteq \mathbb{R}^{3} $$
Do you think the statement \((\mathbb{R}-\mathbb{Z}) \times \mathbb{N}=(\mathbb{R} \times \mathbb{N})-(\mathbb{Z} \times \mathbb{N})\) is true, or false? Justify.
Sketch the following sets of points in the \(x-y\) plane. $$ \left\\{(x, y): x, y \in \mathbb{R}, x^{2}+y^{2} \leq 1\right\\} $$
Sketch the set \(X=[-1,3] \times[0,2]\) on the plane \(\mathbb{R}^{2}\). On separate drawings, shade in the sets \(\bar{X}\) and \(\bar{X} \cap([-2,4] \times[-1,3])\)
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