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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![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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd5b21877-0cf5-4f19-8488-f7f5166e6141%2F723eb2bd-75fa-4748-a28d-4ba73b4edcc4%2Fguq0y8f_processed.jpeg&w=3840&q=75)
Transcribed Image Text: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
Transcribed Image Text: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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