**Problem 5: Category of Relations**Prove that there exists a category \( \text{Rel} \) where the objects are sets, and the morphisms are relations between these sets, with composition defined by relation composition. Specifically, establish the following:(a) For any sets \( A \) and \( B \), and for any relation \( R: A \rightrightarrows B \), demonstrate that:\[\Delta_A ; R = R \quad \text{and} \quad R ; \Delta_B = R,\]where \( \Delta_A: A \rightrightarrows A \) and \( \Delta_B: B \rightrightarrows B \) represent the identity relations on \( A \) and \( B \) respectively.(b) Given any three relations:\[A \xrightarrow{\ \ R\ \ } B \xrightarrow{\ \ S\ \ } C \xrightarrow{\ \ T\ \ } D,\]prove the equality:\[(R ; S) ; T = R ; (S ; T).\]This problem guides the construction and verification of the structure of a category where sets are considered objects, and their interrelations serve as morphisms. It emphasizes the associative property within the context of the composition of relations.

Answered: Image /qna-images/answer/cf2b361d-ec0c-4970-af74-ae8224846c89.jpg

5. Show that there is a category Rel whose objects are the sets and whose morphisms are the relations between sets, with composition given by relation composition. That is to say, prove the following: (a) For every pair of sets A and B and every relation R: A --→ B, we have that AA; R = R and R; AB = R, where AA: A -- A and AB: B --→ B are the identity relations on A and B, respectively. (b) Given any three relations RBSC - D we have the equality (R; S); T = R; (S; T).

5. Show that there is a category Rel whose objects are the sets and whose morphisms are the relations between sets, with composition given by relation composition. That is to say, prove the following: (a) For every pair of sets A and B and every relation R: A --→ B, we have that AA; R = R and R; AB = R, where AA: A -- A and AB: B --→ B are the identity relations on A and B, respectively. (b) Given any three relations RBSC - D we have the equality (R; S); T = R; (S; T).

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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