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.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.7: Relations
Problem 6TFE: Label each of the following statements as either true or false. Let R be a relation on a nonempty...
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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.
By definition of R-1 and since x R¯¹ y, then y R x.
1
By definition of R-¹ and since x R y, then
1
y
R¹ x.
X.
By definition of R¹ and since y R¯¹ x, then x R y.
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.
1
2. ---Select---
V
3. Since R is symmetric and y R x, then x Ry.
4. ---Select---
5. Therefore, by definition of a symmetric relation, R=¹ is symmetric.
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