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 EN= {1,2,3,...}. Here is a step-by-step guide: %3D

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

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section: Chapter Questions
Problem 63RE
See similar textbooks
icon
Related questions
Question
3. 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: (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' = RCRdoes not even depend on
Transcribed Image Text:3. 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: (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' = RCRdoes not even depend on
whether R is transitive or not. Thus, the base case holds vacuously. For the inductive case, suppose that RK C Runder 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 A. Thus, suppose that (a, c) E Rk+1 = Rk o R. This means that there exists b e A such that (a, b) E R and (b, c) E R*. Explain briefly why you can conclude that (a, c) e R also, hence showing that A 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 is, in fact, very easy to establish. For, let (a, b), (b, c) E R. Conclude that (a, c) E R from the assumption that R" CR 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 Runder 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 A. Thus, suppose that (a, c) E Rk+1 = Rk o R. This means that there exists b e A such that (a, b) E R and (b, c) E R*. Explain briefly why you can conclude that (a, c) e R also, hence showing that A 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 is, in fact, very easy to establish. For, let (a, b), (b, c) E R. Conclude that (a, c) E R from the assumption that R" CR 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 2 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Elements Of Modern Algebra
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,
College Algebra
College Algebra
Algebra
ISBN:
9781938168383
Author:
Jay Abramson
Publisher:
OpenStax
Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning
SEE MORE TEXTBOOKS