Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
In this exercise, you want to show that a relation RC A? is transitive + R"C R for all n E N = {1,2, 3, .}. Here is a step-by-step guide:

In this exercise, you want to show that a relation RC A? is transitive + R"C R for all n E N = {1,2, 3, .}. Here is a step-by-step guide:

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 17EQ
See similar textbooks
icon
Related questions
Question
3. In this exercise, you want to show that a relation R C A² is transitive + R" C R for all n E N = {1,2,3, .}. Here is a step-by-step guide: (a) We are going to establish the direction first. We accomplish this by induction on n. For the base case, we have n = 1, but R' = RC R does not even depend on %3D
Transcribed Image Text:3. In this exercise, you want to show that a relation R C A² is transitive + R" C R for all n E N = {1,2,3, .}. Here is a step-by-step guide: (a) We are going to establish the direction first. We accomplish this by induction on n. For the base case, we have n = 1, but R' = RC R does not even depend on %3D
whether R is transitive or not. Thus, the base case holds vacuously. For the inductive case, suppose that Rk C R under the assumption that R is transitive. We want to show that Rk+1 is also a subset or R. To do this, we want to show that if (a, c) E Rk+1, then (a, c) E R also-call this property . Thus, suppose that (a, c) E R*+1 = R* o R. This means that there exists b e A such that (a, b) E R and (b, c) € Rk. Explain briefly why you can conclude that (a, c) E R also, hence showing that holds. (Hint: You are using induction for the proof.) (b) We have established the direction = above; we now want to establish the converse direction +. This direction in fact, very easy to establish. For, let (a, b), (b, c) E R. Conclude that (a, c) € R from the assumption that R" C R for all n e N. (Hint: Use Universal Instantiation on “R" C R for all n E N," specifying the precise value of n that makes the proof work.)
Transcribed Image Text:whether R is transitive or not. Thus, the base case holds vacuously. For the inductive case, suppose that Rk C R under the assumption that R is transitive. We want to show that Rk+1 is also a subset or R. To do this, we want to show that if (a, c) E Rk+1, then (a, c) E R also-call this property . Thus, suppose that (a, c) E R*+1 = R* o R. This means that there exists b e A such that (a, b) E R and (b, c) € Rk. Explain briefly why you can conclude that (a, c) E R also, hence showing that holds. (Hint: You are using induction for the proof.) (b) We have established the direction = above; we now want to establish the converse direction +. This direction in fact, very easy to establish. For, let (a, b), (b, c) E R. Conclude that (a, c) € R from the assumption that R" C R for all n e N. (Hint: Use Universal Instantiation on “R" C R for all n E N," specifying the precise value of n that makes the proof work.)
Expert Solution
steps

Step by step

Solved in 3 steps with 3 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Elements Of Modern Algebra
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,
Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning
SEE MORE TEXTBOOKS