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Prove the following statement. If R is any symmetric relation on a set A, then R -1 is symmetric. Some of the sentences in the following scrambled list can be used in a proof. By definition of R and since x R-1 y, then y Rx. By definition of R-1 and since x R y, then y RT 1 By definition of R-1 and since y R-1 x, then x R y. By definition of R-1 and since y R x, then x RT -1 y. 3. Since R is symmetric and y R x, then x R y. 4. ---Select--- X. Proof: We construct a proof by selecting appropriate sentences from the list and inserting them in the spaces. Construct the proof by using the appropriate statements from the list and putting them in the correct order. 1. Let R be any symmetric relation on a set A, and suppose that x and y are any elements of A such that x R¹ y. 2. ---Select--- 5. Therefore, by definition of a symmetric relation, R 1 is symmetric.

Prove the following statement. If R is any symmetric relation on a set A, then R -1 is symmetric. Some of the sentences in the following scrambled list can be used in a proof. By definition of R and since x R-1 y, then y Rx. By definition of R-1 and since x R y, then y RT 1 By definition of R-1 and since y R-1 x, then x R y. By definition of R-1 and since y R x, then x RT -1 y. 3. Since R is symmetric and y R x, then x R y. 4. ---Select--- X. Proof: We construct a proof by selecting appropriate sentences from the list and inserting them in the spaces. Construct the proof by using the appropriate statements from the list and putting them in the correct order. 1. Let R be any symmetric relation on a set A, and suppose that x and y are any elements of A such that x R¹ y. 2. ---Select--- 5. Therefore, by definition of a symmetric relation, R 1 is symmetric.

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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Prove the following statement. If \( R \) is any symmetric relation on a set \( A \), then \( R^{-1} \) is symmetric. Some of the sentences in the following scrambled list can be used in a proof. 1. By definition of \( R^{-1} \) and since \( x \, R^{-1} \, y \), then \( y \, R \, x \). 2. By definition of \( R^{-1} \) and since \( x \, R \, y \), then \( y \, R^{-1} \, x \). 3. By definition of \( R^{-1} \) and since \( y \, R^{-1} \, x \), then \( x \, R \, y \). 4. By definition of \( R^{-1} \) and since \( y \, R \, x \), then \( x \, R^{-1} \, y \). **Proof:** We construct a proof by selecting appropriate sentences from the list and inserting them in the spaces. Construct the proof by using the appropriate statements from the list and putting them in the correct order. 1. Let \( R \) be any symmetric relation on a set \( A \), and suppose that \( x \) and \( y \) are any elements of \( A \) such that \( x \, R^{-1} \, y \). 2. ---Select--- 3. Since \( R \) is symmetric and \( y \, R \, x \), then \( x \, R \, y \). 4. ---Select--- 5. Therefore, by definition of a symmetric relation, \( R^{-1} \) is symmetric.
Transcribed Image Text:Prove the following statement. If \( R \) is any symmetric relation on a set \( A \), then \( R^{-1} \) is symmetric. Some of the sentences in the following scrambled list can be used in a proof. 1. By definition of \( R^{-1} \) and since \( x \, R^{-1} \, y \), then \( y \, R \, x \). 2. By definition of \( R^{-1} \) and since \( x \, R \, y \), then \( y \, R^{-1} \, x \). 3. By definition of \( R^{-1} \) and since \( y \, R^{-1} \, x \), then \( x \, R \, y \). 4. By definition of \( R^{-1} \) and since \( y \, R \, x \), then \( x \, R^{-1} \, y \). **Proof:** We construct a proof by selecting appropriate sentences from the list and inserting them in the spaces. Construct the proof by using the appropriate statements from the list and putting them in the correct order. 1. Let \( R \) be any symmetric relation on a set \( A \), and suppose that \( x \) and \( y \) are any elements of \( A \) such that \( x \, R^{-1} \, y \). 2. ---Select--- 3. Since \( R \) is symmetric and \( y \, R \, x \), then \( x \, R \, y \). 4. ---Select--- 5. Therefore, by definition of a symmetric relation, \( R^{-1} \) is symmetric.
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