Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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**Question:** Is the set \( X \) of all functions \( f : \mathbb{Z}^+ \rightarrow \{0,1\} \) countable? Prove or disprove. --- To determine if the set \( X \) of all functions from the set of positive integers \( \mathbb{Z}^+ \) to the set \(\{0,1\} \) is countable, we first need to understand some key concepts: 1. **Countable Sets:** - A set is **countably infinite** if its elements can be placed in one-to-one correspondence with the positive integers \( \mathbb{Z}^+ \) (i.e., it can be listed as \( \{a_1, a_2, a_3, \ldots\} \)). - A set is **uncountable** if it is not countable (i.e., it cannot be listed in this manner). 2. **Functions:** - A function \( f : \mathbb{Z}^+ \rightarrow \{0,1\} \) assigns to each positive integer either 0 or 1. - Each function can be represented as an infinite sequence of 0s and 1s. For example, one such function might look like \( f(1)=0, f(2)=1, f(3)=0, \ldots \), which corresponds to the sequence \( (0, 1, 0, \ldots) \). 3. **Cardinality:** - The set of all infinite sequences of 0s and 1s has the same cardinality as the power set of the natural numbers \( \mathcal{P}(\mathbb{N}) \). - The cardinality of \( \mathcal{P}(\mathbb{N}) \) is \( 2^{\aleph_0} \) (read as \( 2 \) raised to the power of \( \aleph_0 \), where \( \aleph_0 \) denotes the cardinality of the set of positive integers \( \mathbb{Z}^+ \), representing the smallest infinity, also known as countable infinity). **Proof:** 1. Assume \( X \) is countable. Then, there would exist a bijection between the set of all functions \(
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Transcribed Image Text:**Question:** Is the set \( X \) of all functions \( f : \mathbb{Z}^+ \rightarrow \{0,1\} \) countable? Prove or disprove. --- To determine if the set \( X \) of all functions from the set of positive integers \( \mathbb{Z}^+ \) to the set \(\{0,1\} \) is countable, we first need to understand some key concepts: 1. **Countable Sets:** - A set is **countably infinite** if its elements can be placed in one-to-one correspondence with the positive integers \( \mathbb{Z}^+ \) (i.e., it can be listed as \( \{a_1, a_2, a_3, \ldots\} \)). - A set is **uncountable** if it is not countable (i.e., it cannot be listed in this manner). 2. **Functions:** - A function \( f : \mathbb{Z}^+ \rightarrow \{0,1\} \) assigns to each positive integer either 0 or 1. - Each function can be represented as an infinite sequence of 0s and 1s. For example, one such function might look like \( f(1)=0, f(2)=1, f(3)=0, \ldots \), which corresponds to the sequence \( (0, 1, 0, \ldots) \). 3. **Cardinality:** - The set of all infinite sequences of 0s and 1s has the same cardinality as the power set of the natural numbers \( \mathcal{P}(\mathbb{N}) \). - The cardinality of \( \mathcal{P}(\mathbb{N}) \) is \( 2^{\aleph_0} \) (read as \( 2 \) raised to the power of \( \aleph_0 \), where \( \aleph_0 \) denotes the cardinality of the set of positive integers \( \mathbb{Z}^+ \), representing the smallest infinity, also known as countable infinity). **Proof:** 1. Assume \( X \) is countable. Then, there would exist a bijection between the set of all functions \(
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