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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).

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.7: Relations
Problem 14E: In each of the following parts, a relation is defined on the set of all human beings. Determine...
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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 R --→ AA; R = R and R; AB = R, where ΔΑ: A -- A and AB: B --→ B are the identity relations on A and B, respectively. (b) Given any three relations --> B we have the equality (R; S) ; T = R ; (S ; T). B, we have that S -- C ---- D
Transcribed Image Text: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 R --→ AA; R = R and R; AB = R, where ΔΑ: A -- A and AB: B --→ B are the identity relations on A and B, respectively. (b) Given any three relations --> B we have the equality (R; S) ; T = R ; (S ; T). B, we have that S -- C ---- D
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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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