Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
Let O be the set of all odd integers, and let 2Z be the set of all even integers. Prove that O has the same cardinality as 2z. Proof: In order to show that O has the same cardinality as 22 we must show that there is a well-defined function f: 0 -→ 2Z that is both one-to-one and onto. We will show that the following is a function from 0 to 2Z that satisfies these requirements. (Choose one definition for f and use it for the rest of the proof.) O f(n) = ", for each odd integer n in o O f(n) = Inl, for each odd integer n in o O f(n) = n - 1, for each odd integer n in o O f(n) = n+ 2, for each odd integer n in o O f(n) - 3n, for each odd integer n in o Well-Defined One-to-One Onto Proof that f is onto: To show that f is onto, let m be any even integer in 2z. By definition of even, there exists an integer k such that m = 2k. On a separate piece of scratch paper, find an odd integer in 0, written as an expression using the variable k, with the property that when fis applied to it, the result is 2k. Write the expression in the box below. By construction, the quantity in the box is an odd integer, and when the function fis applied to it, the result is the even integer 2k, which equals m. Thus, we have shown that there exists an element of O that is sent to m by f, and so f is onto. Conclusion: Since f is a well-defined function from O to 2Z that is one-to-one and onto, we conclude that O and 22 have the same cardinality.

Let O be the set of all odd integers, and let 2Z be the set of all even integers. Prove that O has the same cardinality as 2z. Proof: In order to show that O has the same cardinality as 22 we must show that there is a well-defined function f: 0 -→ 2Z that is both one-to-one and onto. We will show that the following is a function from 0 to 2Z that satisfies these requirements. (Choose one definition for f and use it for the rest of the proof.) O f(n) = ", for each odd integer n in o O f(n) = Inl, for each odd integer n in o O f(n) = n - 1, for each odd integer n in o O f(n) = n+ 2, for each odd integer n in o O f(n) - 3n, for each odd integer n in o Well-Defined One-to-One Onto Proof that f is onto: To show that f is onto, let m be any even integer in 2z. By definition of even, there exists an integer k such that m = 2k. On a separate piece of scratch paper, find an odd integer in 0, written as an expression using the variable k, with the property that when fis applied to it, the result is 2k. Write the expression in the box below. By construction, the quantity in the box is an odd integer, and when the function fis applied to it, the result is the even integer 2k, which equals m. Thus, we have shown that there exists an element of O that is sent to m by f, and so f is onto. Conclusion: Since f is a well-defined function from O to 2Z that is one-to-one and onto, we conclude that O and 22 have the same cardinality.

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
See similar textbooks
icon
Related questions
Question

Proof that f is onto

**Title: Proving the Cardinality Equivalence between Odd and Even Integers** **Objective:** Demonstrate that the set \( O \) of all odd integers and the set \( 2\mathbb{Z} \) of all even integers have the same cardinality. **Proof Outline:** To prove that \( O \) and \( 2\mathbb{Z} \) have the same cardinality, we must establish a one-to-one and onto (bijective) function \( f: O \to 2\mathbb{Z} \). ### Function Definition: We select the function \( f(n) = n - 1 \) for each odd integer \( n \) in \( O \). **Rationale:** This function will be used to establish the required properties (well-defined, one-to-one, and onto) for the bijection. --- **Proof that \( f \) is Onto:** 1. **Target:** Let \( m \) be an arbitrary even integer in \( 2\mathbb{Z} \). By definition, there exists an integer \( k \) such that \( m = 2k \). 2. **Finding an Odd Integer:** Our task is to find an odd integer \( n \) in \( O \) such that when \( f(n) \) is applied, the result will be \( 2k \). 3. **Expression:** Note down the odd integer in the box provided as \( n = 2k + 1 \). 4. **Verification:** When \( f \) is applied to \( n \), i.e., \( f(n) = (2k + 1) - 1 = 2k \), which is equal to \( m \). Thus, every even integer \( m \) can be represented as \( f(n) \), showing \( f \) is onto. **Conclusion:** Since \( f \) is well-defined, one-to-one, and onto, we conclude that the odd integers \( O \) and even integers \( 2\mathbb{Z} \) have the same cardinality.
Transcribed Image Text:**Title: Proving the Cardinality Equivalence between Odd and Even Integers** **Objective:** Demonstrate that the set \( O \) of all odd integers and the set \( 2\mathbb{Z} \) of all even integers have the same cardinality. **Proof Outline:** To prove that \( O \) and \( 2\mathbb{Z} \) have the same cardinality, we must establish a one-to-one and onto (bijective) function \( f: O \to 2\mathbb{Z} \). ### Function Definition: We select the function \( f(n) = n - 1 \) for each odd integer \( n \) in \( O \). **Rationale:** This function will be used to establish the required properties (well-defined, one-to-one, and onto) for the bijection. --- **Proof that \( f \) is Onto:** 1. **Target:** Let \( m \) be an arbitrary even integer in \( 2\mathbb{Z} \). By definition, there exists an integer \( k \) such that \( m = 2k \). 2. **Finding an Odd Integer:** Our task is to find an odd integer \( n \) in \( O \) such that when \( f(n) \) is applied, the result will be \( 2k \). 3. **Expression:** Note down the odd integer in the box provided as \( n = 2k + 1 \). 4. **Verification:** When \( f \) is applied to \( n \), i.e., \( f(n) = (2k + 1) - 1 = 2k \), which is equal to \( m \). Thus, every even integer \( m \) can be represented as \( f(n) \), showing \( f \) is onto. **Conclusion:** Since \( f \) is well-defined, one-to-one, and onto, we conclude that the odd integers \( O \) and even integers \( 2\mathbb{Z} \) have the same cardinality.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Follow-up Questions
Read through expert solutions to related follow-up questions below.
Follow-up Question

Find an odd integer o written as an expression using the variable k with the property that when f is applied to it the result is 2k

Solution
Bartleby Expert
SEE SOLUTION
Knowledge Booster
Complexity
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS