1.9
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Suppose there exist three functions \(f: A \xrightarrow{1-1} B\), \(g: B \xrightarrow{1-1} C\), and \(h: C \xrightarrow{1-1} A\). Prove that \(A \approx B \approx C\).
- \(A \approx B\)
I will use Theorem C-S-B to prove that there must exist a bijection between \(A\) and \(B\)
Thus \(\exists f: A \to B\) injective.
\(r = h \circ g: B \to A\) is injective because it is the composition of two injective functions. * \(C \approx A\)
\(h: C \to A\) injective.
\(s = g \circ f\) is injective because it is the composition of two injective functions
Thus \(s: A \to C\).
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If possible, give an example of:
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Functions \(f\) and \(g\) such that \(f: \mathbb{Q} \xrightarrow{1-1} \mathbb{N}\) and \(g: \mathbb{N} \xrightarrow{1-1} \mathbb{Q}\), but neither \(f\) nor \(g\) are surjective.
\(g: \mathbb{N} \xrightarrow{1-1} \mathbb{Q}\) where \(n\mapsto n\). This is not surjective because \(\frac{1}{2} \notin \text{Im}(g)\).
\(f: \mathbb{Q} \xrightarrow{1-1} \mathbb{N}\). For \(\frac{a}{b} \in \mathbb{Q}\) (with \(\gcd(a, b) = 1\)):
- If \(a > 0\), map \(\frac{a}{b} \mapsto 2^a 3^b\).
- If \(a < 0\), map \(\frac{a}{b} \mapsto 5^{-a} 7^b\).
This is not surjective because \(13 \notin \text{Im}(f)\). 2. A function \(f: \mathbb{R} \xrightarrow{1-1} \mathbb{N}\).
Impossible 3. A function \(f: \mathcal{P}(\mathbb{N}) \xrightarrow{1-1} \mathbb{N}\).
Impossible 4. A function \(f: \mathbb{R} \xrightarrow{1-1} \mathbb{Q}\).
Impossible
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Prove that \(\mathbb{R}^2 \approx \mathbb{R}\)
I will try to find two injections, then use the C-S-B Theorem.
\(f: \mathbb{R} \to \mathbb{R}^2\) where \(x \mapsto (x, 1)\).
\(g: \mathbb{R}^2 \xrightarrow{1-1} \mathbb{R}\)
We know that \(\mathbb{R} \approx (0, 1)\).
\(g_2: (0, 1) \times (0, 1) \to (0, 1)\) is defined as follows \((x, y) \mapsto 0.x_1 y_1 x_2 y_2 x_3 y_3 \dots\)
But we need to express as infinite decimal number such as \(0.1\to0.09999...\)
And it is not surjective since \(0.1010101 \dots\) and \(0.1101101 \dots\) can not be expressed