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Use definition of set X has the same cardinality as set Y prove that Card(R = {2}) = Card(R – {0}).

Use definition of set X has the same cardinality as set Y prove that Card(R = {2}) = Card(R – {0}).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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### Problem Statement **3.** Use the definition of set \( X \) having the same cardinality as set \( Y \) to prove that \( \text{Card}(R - \{2\}) = \text{Card}(R - \{0\}) \). ### Explanation 1. **Definition of Cardinality:** The cardinality of a set \( A \), denoted \( \text{Card}(A) \), is a measure of the "number of elements" in the set. Two sets \( X \) and \( Y \) are said to have the same cardinality if there exists a bijective function (one-to-one and onto) between them. 2. **Given Sets:** - \( R \) is the set of all real numbers. - \( R - \{2\} \) represents the set of all real numbers except 2. - \( R - \{0\} \) represents the set of all real numbers except 0. ### Proof Outline To prove that \( \text{Card}(R - \{2\}) = \text{Card}(R - \{0\}) \), we need to establish a bijective function between the sets \( R - \{2\} \) and \( R - \{0\} \). ### Proof 1. **Define a Function:** We define a function \( f: R - \{2\} \rightarrow R - \{0\} \) by \( f(x) = \frac{x - 2}{x} \). This function takes a real number \( x \) (except 2) and maps it to another real number \( (except 0) \). 2. **Prove Injectivity:** Let \( f(x_1) = f(x_2) \). This means \( \frac{x_1 - 2}{x_1} = \frac{x_2 - 2}{x_2} \). Simplifying, we get: \( x_1(x_2 - 2) = x_2(x_1 - 2) \). This simplifies further to: \( x_1 x_2 - 2x_1 = x_2 x_1 - 2x_2 \). Canceling out \( x_1 x_2 \)
Transcribed Image Text:### Problem Statement **3.** Use the definition of set \( X \) having the same cardinality as set \( Y \) to prove that \( \text{Card}(R - \{2\}) = \text{Card}(R - \{0\}) \). ### Explanation 1. **Definition of Cardinality:** The cardinality of a set \( A \), denoted \( \text{Card}(A) \), is a measure of the "number of elements" in the set. Two sets \( X \) and \( Y \) are said to have the same cardinality if there exists a bijective function (one-to-one and onto) between them. 2. **Given Sets:** - \( R \) is the set of all real numbers. - \( R - \{2\} \) represents the set of all real numbers except 2. - \( R - \{0\} \) represents the set of all real numbers except 0. ### Proof Outline To prove that \( \text{Card}(R - \{2\}) = \text{Card}(R - \{0\}) \), we need to establish a bijective function between the sets \( R - \{2\} \) and \( R - \{0\} \). ### Proof 1. **Define a Function:** We define a function \( f: R - \{2\} \rightarrow R - \{0\} \) by \( f(x) = \frac{x - 2}{x} \). This function takes a real number \( x \) (except 2) and maps it to another real number \( (except 0) \). 2. **Prove Injectivity:** Let \( f(x_1) = f(x_2) \). This means \( \frac{x_1 - 2}{x_1} = \frac{x_2 - 2}{x_2} \). Simplifying, we get: \( x_1(x_2 - 2) = x_2(x_1 - 2) \). This simplifies further to: \( x_1 x_2 - 2x_1 = x_2 x_1 - 2x_2 \). Canceling out \( x_1 x_2 \)
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