Please answer questions 1-10, If possible.When answering these questions, please conduct the following:Summarize your work on each of the questions. Support your conclusions with evidence.Specifically address questions #8 and #10.Explain how this exercise can be used to “create integers entirely from the whole numbers.” 1. The whole numbers {0, 1, 2, .} with addition do not form a group. Why?Think of the set of all ordered pairs of whole numbers, such as (2, 5). The operation + on ordered pairs isdefined as follows: (a, b) + (c, d) = (a + c, b + d)2. Calculate:а. (3,3) + (5,6)b. (4,3) + (5,6)c. (2,5) + (4,2)Say that the two pairs (a, b) and (c, d) are equivalent when a + d == b+c.a. Name some pairs that are equivalent to (2,5).b. Display them on a graph.3.4. Repeat #3 for (2,2) and (4,2), making sure that each equivalence class is easy to distinguishfrom the others on the graph.5. What is the smallest pair that is equivalent to (5,9)? To (9,5)?6. Choose two equivalence classes, A and B. For example, you might choose the class of (2,5) for Aand the class of (4,2) for B. Add a number pair from A to a number pair from B. Try it again withother pairs from A and B. Are the results equivalent? Explain algebraically why they must beequivalent. 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 questions 1-10, If possible.When answering these questio
Uploaded: 2024-03-20T06:38:44.000Z
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Description: Please answer questions 1-10, If possible.When answering these questions, please conduct the following:Summarize your work on each of the questions. Support your conclusions with evidence.Specifically address questions #8 and #10.Explain how this exe