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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![**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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdf527488-7949-457b-8fb2-a06535c2214c%2Fcf2b361d-ec0c-4970-af74-ae8224846c89%2Fb1t7qo_processed.png&w=3840&q=75)
Transcribed Image Text:**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.
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Hello, were you able to answer both parts a and b? I am kind of confused about which part is what.
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