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 {TOlisa group.8. Matsh ihe elements in T with the integers and show that TO) has the same structure astheintegers with edaition.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 Shew that the definitien of Omakes 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

The multiplication for: Take, . Then, For a, b = 10, 2 and c, d =2, 10 Then, a, b.c, d = 10, 2.2, 10=100+4, 20-20= 104,0 For a, b = 5, 6 and c, d =5, 6 Then, a, b.c, d = 5, 6.5, 6=30+30, 25-36= 60,-11 So, a, b.c, d = ad +bc, ac -bd is nothing but T,⊗. Hence, T,⊗ has the same structure as the integers with multiplication.

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Let I be the set of equivalence classes as defined as above. Define O as follows: if A and B are equivalence classes in T, A O B is the equivalence class of the sum of an element from A and an element from B. For example, using the A and B from #6: (2,5) + (4,2) = (6,7), so, A Ð B is the equivalence class of (6,7): {(0,1), (1,2), (2,3), ...} Đis well defined because as we showed in #6, the result of the operation does not depend on which representatives of the equivalence classes we choose.

Let I be the set of equivalence classes as defined as above. Define O as follows: if A and B are equivalence classes in T, A O B is the equivalence class of the sum of an element from A and an element from B. For example, using the A and B from #6: (2,5) + (4,2) = (6,7), so, A Ð B is the equivalence class of (6,7): {(0,1), (1,2), (2,3), ...} Đis well defined because as we showed in #6, the result of the operation does not depend on which representatives of the equivalence classes we choose.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 11E: Let be a relation defined on the set of all integers by if and only if sum of and is odd. Decide...

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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 are
equivalence classes in T, A O B is the equivalence class of the sum of an element from A and an
element from B. For example, using the A and B from #6: (2,5) + (4,2) = (6,7), so, A O B is the
equivalence class of (6,7): {(0,1), (1,2), (2,3), ...} Đis well defined because as we showed in #6, the
result of the operation does not depend on which representatives of the equivalence classes we choose.
7. Show that {TOlisa group.
8. Matsh ihe elements in T with the integers and show that TO) has the same structure asthe
integers with edaition.
Multiplication on the number pairs is defined as follows: (a, b) · (c, d) = (ad + bc, ac + bd).O in T is
defined in a similar way to O above from multiplication of number pairs.
9 Shew that the definitien of Omakes sense.
10. Perhaps surprisingly, {T,8} has the same structure as the integers with multiplication. Check
that on some examples. Why does this work?

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