8. We defined the standard order and the divisibility order | for natural numbers from the operations + (whose identity is 0) and (whose identity is 1), respectively, using very similar definitions and proved that they satisfied similar properties. We can do it in a more abstract setting following exactly the same ideas: Suppose that we are given a set X together with an operation * on X and a particular element e E X satisfying the following properties: the operation is associative: that is, for all a, b, c E X, we have a* (b* c) = (a*b) * c. e is an identity for *: that is, for every a E X, a *e = a and e* a = a. Se define the relation as follows: for all a, b E X, we have that ab if and only if a* d = b for some d E X.

8. We defined the standard order and the divisibility order | for natural numbers from the operations + (whose identity is 0) and (whose identity is 1), respectively, using very similar definitions and proved that they satisfied similar properties. We can do it in a more abstract setting following exactly the same ideas: Suppose that we are given a set X together with an operation * on X and a particular element e E X satisfying the following properties: the operation is associative: that is, for all a, b, c E X, we have a* (b* c) = (ab) c. e is an identity for : that is, for every a E X, a e = a and e* a = a. Se define the relation as follows: for all a, b E X, we have that ab if and only if a* d = b for some d E X.

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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