Please answer questions 1-10.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 is defined as follows: (a, b) + (c, d) = (a+c, b+d).2. **Calculate:** a. (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.3. **a.** Name some pairs that are equivalent to (2, 5). **b.** Display them on a graph. The set of all pairs that are equivalent to (2, 5) is called an **equivalence class**.4. **Repeat #3 for (2, 2) and (4, 2),** making sure that each equivalence class is easy to distinguish from 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 A and the class of (4, 2) for B. Add a number pair from A to a number pair from B. Try it again with other pairs from A and B. Are the results equivalent? Explain algebraically why they must be equivalent. The image contains text related to mathematical concepts involving equivalence classes, groups, and operations. Here's the transcription and explanation:---Let T be the set of equivalence classes as defined as above. Define ⊕ as follows: if A and B are equivalence classes in T, A⊕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⊕B is the equivalence class of (6,7): {(0,1), (1,2), (2,3), …} ⊕ is 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 {T, ⊕} is a group.8. Match the elements in T with the integers, and show that {T, ⊕} has the same structure as the integers with addition.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 ⊕ above from multiplication of number pairs.9. Show that the definition of ⊗ makes sense.10. Perhaps surprisingly, {T, ⊗} has the same structure as the integers with multiplication. Check that on some examples. Why does this work?This approach is a way to create integers entirely from the whole numbers.---The text provides a detailed explanation of operations on equivalence classes to form structures analogous to integers, defining both addition (⊕) and multiplication (⊗) operations for elements in set T. The goal is to demonstrate that these operations form a group and are structurally similar to integer operations.

Name: Please answer questions 1-10.Specifically, address questions #8 and #1
Uploaded: 2024-03-05T11:53:49.000Z
Channel: Bartleby
Description: Please answer questions 1-10.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