### Problem Statement:**9.** Suppose \( B \subseteq A \), \( B \neq A \), and \( B \sim A \). Prove that \( A \) is infinite.---### Explanation and Solution:**Given:**- \( B \subseteq A \): Set \( B \) is a subset of set \( A \).- \( B \neq A \): Set \( B \) is not equal to set \( A \); hence, \( B \) is a proper subset of \( A \).- \( B \sim A \): Set \( B \) is equivalent to set \( A \), meaning there exists a bijection (one-to-one and onto function) between \( B \) and \( A \).**Objective:**- Prove that set \( A \) is infinite.**Proof:**To prove that \( A \) is infinite, assume that \( A \) is finite and derive a contradiction.1. **Assumption (for contradiction)**: Suppose \( A \) is finite. - This implies that \( A \) has a finite number of elements, say \( n \).2. **Proper subset implication**: - Since \( B \subseteq A \) and \( B \neq A \), \( B \) must have fewer elements than \( A \), i.e., the number of elements in \( B \) is less than \( n \).3. **Cardinality argument**: - If \( B \) is a proper subset of \( A \) and \( A \) is finite, the number of elements in \( B \) must be \( m \), where \( 0 \leq m < n \).4. **Contradiction with equivalence**: - Given \( B \sim A \), there exists a bijection between \( B \) and \( A \). - For finite sets, a bijection requires both sets to have the same number of elements for one-to-one correspondence. - Hence, \( B \) should have \( n \) elements if it is equivalent to \( A \), but \( B \) has only \( m \) elements (\( m < n \)), which is a contradiction.5. **Conclusion**: - Therefore, the assumption that set \( A \) is finite leads to a contradiction. - Hence, \( A \) must

Name: ### Problem Statement:**9.** Suppose \( B \subseteq A \), \( B \neq A
Uploaded: 2024-03-20T06:38:44.000Z
Channel: Bartleby
Description: ### Problem Statement:**9.** Suppose \( B \subseteq A \), \( B \neq A \), and \( B \sim A \). Prove that \( A \) is infinite.---### Explanation and Solution:**Given:**- \( B \subseteq A \): Set \( B \) is a subset of set \( A \).- \( B \neq A \): Se

Answered: Image /qna-images/answer/39bc91cd-81a9-4197-ba7f-f1c7b9caefc7.jpg

Answered: Suppose B C A, B + A, and B ~ A. Prove…

Math

Advanced Math

Suppose B C A, B + A, and B ~ A. Prove that A is infinite.

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

Similar questions

1. Let a, b e Z so that 2a + 3b is a multiple of 11. Prove that 11 divides a - 4b.
Find all incongruent quadratic residues and quadratic nonresidues modulo 59. How many of each are there?
Let [A1, A2], [B1, B2], [C1, C2] be 3 married couples who attended a concert. Show that the number of ways to seat the 3 married couples attending a concert is equal to 72 if and only if members of the same sex are all seated next to each other in the 6 row seats?
Q1) Compute the products using the definition as a linear combination. 8 3 [5 1 -4 2
Suppose a, b € R and a < b. Prove that |R| = |(a,b)| = |[a, b)| = |(a,b]| = |[a, b]|.
Discrete Math.
Q11: If X and Y are denumerable sets, then X - Y is countable.
1. Show that the set of integers S = {a² + b² : a, b € Z} that can be respresented as the sum of two squares is closed under multiplication. Hint: (a? + b)(c? + d²) = N(a + bi)N(c + di) = N[(a + bi)(c + di)] = - ...
4. Show that: TT de ib//(a? – b?) a+bcose b2 Where a>b>0 such that * -1+ 1 < 1 a²
Ex 2. prove: - a ≤a for every real number a .
Only Need Part(d), thank you!