Consider the relation~ defined on Zx N = {(x,y) : x € Z, and y € N} by (a,b)~ (c,d) ad=bc. (a) Prove that is an equivalence relation. (b) List several elements of the equivalence class of (2,3). Repeat for the equivalence class of (-3,7). What do the equivalence classes have to do with the set of rational numbers Q? (c) Define operations and on Zx N by [(a,b)] → [(c,d)] = [(ad + bc, bd)], [(a, b)] [(c,d)]= [(ac, bd)]. Prove that and are well-defined. Try to do this question without using division! We will return to this example in the next section.

Consider the relation~ defined on Zx N = {(x,y) : x € Z, and y € N} by (a,b)~ (c,d) ad=bc. (a) Prove that is an equivalence relation. (b) List several elements of the equivalence class of (2,3). Repeat for the equivalence class of (-3,7). What do the equivalence classes have to do with the set of rational numbers Q? (c) Define operations and on Zx N by [(a,b)] → [(c,d)] = [(ad + bc, bd)], [(a, b)] [(c,d)]= [(ac, bd)]. Prove that and are well-defined. Try to do this question without using division! We will return to this example in the next section.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Problem 4 needed Read the instructions fron one pic and read the statement 2nd picture Please provide me hundred percent efficient solution in the order to get positive feedback

7.5.4 Consider the relation~ defined on Zx N = {(x,y) : x = Z, and y N} by
(a,b)~ (c, d)
ad=bc.
(a) Prove that is an equivalence relation.
(b) List several elements of the equivalence class of (2,3). Repeat for the equivalence class of
(-3,7). What do the equivalence classes have to do with the set of rational numbers Q?
ZxN
(c) Define operations and on
xN/by
[(a, b)] [(c,d)] = [(ad + bc, bd)],
Prove that and are well-defined.
Try to do this question without using division! We will return to this example in the next section.
[(a,b)] [(c,d)] = [(ac, bd)].
Ø

4. Problem 7.5.4. You should do this problem without ever using division! Trust the
definitions. The algebra will work out.

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