Answer
See explanation
Work Step by Step
A standard way to visualize the bijection between \((0,1)\) and \(\mathbb{R}\) is via the function
\[
f(x) \;=\; \tan\!\Bigl(\pi x \;-\;\tfrac{\pi}{2}\Bigr),
\quad 0 < x < 1.
\]
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## 1. Understanding the Formula
- Let \(\theta = \pi x - \tfrac{\pi}{2}\).
- As \(x\) goes from \(0\) to \(1\), \(\theta\) goes from \(-\tfrac{\pi}{2}\) to \(+\tfrac{\pi}{2}\).
- The tangent function on \(\bigl(-\tfrac{\pi}{2},+\tfrac{\pi}{2}\bigr)\) takes **all real values** from \(-\infty\) to \(+\infty\).
Hence \(f\) is a continuous, strictly increasing map from \((0,1)\) **onto** \(\mathbb{R}\).
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## 2. Sketch of the Graph
1. **Domain**: \(x \in (0,1)\).
2. **Vertical behavior**:
- As \(x \to 0^+\), \(\theta \to -\tfrac{\pi}{2}^+\), and \(\tan(\theta) \to -\infty\).
- As \(x \to 1^-\), \(\theta \to +\tfrac{\pi}{2}^-\), and \(\tan(\theta) \to +\infty\).
3. **Key midpoint**: When \(x = \tfrac12\), \(\theta = \pi \cdot \tfrac12 - \tfrac{\pi}{2} = 0\), and \(\tan(0) = 0\). So the graph crosses the \(x\)-axis at \(\bigl(\tfrac12,\,0\bigr)\).
4. **Monotonicity**: The function \(\tan(\theta)\) is strictly increasing on \(\bigl(-\tfrac{\pi}{2},+\tfrac{\pi}{2}\bigr)\). Thus \(f(x)\) is strictly increasing from \(-\infty\) (at \(x=0\)) to \(+\infty\) (at \(x=1\)).
Visually, the graph has **vertical asymptotes** at \(x=0\) and \(x=1\) (though these endpoints are not in the domain) and sweeps upward continuously through all real \(y\)-values.
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## 3. Explaining Why \((0,1)\) and \(\mathbb{R}\) Have the Same Cardinality
1. **Onto** \(\mathbb{R}\): For every real number \(y\), there is a unique \(\theta \in \bigl(-\tfrac{\pi}{2},+\tfrac{\pi}{2}\bigr)\) such that \(\tan(\theta) = y\). That \(\theta\) corresponds uniquely to some \(x \in (0,1)\) via \(\theta = \pi x - \tfrac{\pi}{2}\).
2. **One‐to‐one**: The tangent function is strictly increasing on \(\bigl(-\tfrac{\pi}{2},+\tfrac{\pi}{2}\bigr)\), so different \(x\)-values map to different \(y\)-values.
Hence \(f\) is a **bijection** from \((0,1)\) onto \(\mathbb{R}\). Whenever there is a bijection between two sets, they have the same cardinality. Therefore, \(\,(0,1)\) and \(\mathbb{R}\) are **equally infinite** in the sense of cardinality.
