please answer question number 9 Let I be the set of equivalence classes as defined as above. Define O as follows: if A and B areequivalence classes in T, A O B is the equivalence class of the sum of an element from A and anelement from B. For example, using the A and B from #6: (2,5) + (4,2) = (6,7), so, A OB is theequivalence class of (6,7): {(0,1), (1,2), (2,3), ...} Đis well defined because as we showed in #6, theresult of the operation does not depend on which representatives of the equivalence classes we choose.7. Show that {T,O}is a group.8. Match the elements in T with the integers and show that {T,O} has the same structure as theintegers with addition.Multiplication on the number pairs is defined as follows: (a, b) · (c, d) = (ad + bc, ac + bd).O in T isdefined in a similar way to O above from multiplication of number pairs.9. Show that the definition of 8 makes sense.10. Perhaps surprisingly, {T,8} has the same structure as the integers with multiplication. Checkthat on some examples. Why does this work?

Name: please answer question number 9 Let I be the set of equivalence classe
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Description: please answer question number 9 Let I be the set of equivalence classes as defined as above. Define O as follows: if A and B areequivalence classes in T, A O B is the equivalence class of the sum of an element from A and anelement from B. For example

Multiplication on the number pairs is defined as : a, b . c, d =ad+bc , ac+bd ⊗ in T is defined as…

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Multiplication on the number pairs is defined as follows: (a, b) · (c, d) defined in a similar way to O above from multiplication of number pairs. (ad + bc, ac + bd).O in Tis 9. Show that the definition of 8 makes sense.

Multiplication on the number pairs is defined as follows: (a, b) · (c, d) defined in a similar way to O above from multiplication of number pairs. (ad + bc, ac + bd).O in Tis 9. Show that the definition of 8 makes sense.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 3E: a. Let R be the equivalence relation defined on Z in Example 2, and write out the elements of the...

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