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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

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

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.7: Distinguishable Permutations And Combinations
Problem 16E
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Please answer questions 4

When answering these questions, please conduct the following:

  1. Summarize your work on each of the questions. Support your conclusions with evidence.
  2. 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: 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 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.
Transcribed Image Text: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: 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 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.
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