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

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
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
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS