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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