Please help with this question: 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 O B 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 (TO}is a group.SMatch the elemansIwith the integers and show that (TO nas the same structureas 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 O makes sense.10. Perhaps surprisingly, {T,8} has the same structure as the integers with multiplication. Checkthat on some examples. Why does this work?

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Description: Please help with this question: 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

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

Multiplication on the number pairs is defined as follows: (a, b) · (c, d) = (ad + bc, ac + bd).& in T is defined in a similar way to O above from multiplication of number pairs. 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

Chapter5: Rings, Integral Domains, And Fields

Section5.2: Integral Domains And Fields

Problem 7E: [Type here] 7. Let be the set of all ordered pairs of integers and . Equality, addition, and...

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