For the function f: Z18 → Z, defined as f(r) = 4x + 1 mod 6. Consider the relation R on the set Z8defined as a Ry when f(x) = S(y). Show that this relation is:(a) reflexive;(b) symmetric;(c) transitive;this will prove that R is an equivalence relation; finally(d) calculate its equivalence classes.

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Answered: For the function S : Z4s → Za defined…

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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Define an equivalence relation on R² by: ¤1, Y1) ~ (x2, Y2) E if and only if (x1, Y1) and (x2, Y2) have the same distance from the origin. (a) Interpret the equivalence classes geometrically and describe the quotient R?/ ~. (b) Is the function f: (R²/ ~) → R, [(x1,y1)] → xỉ + yỉ well defined?
if f : S →T is a function, then there is a bijection between f(S) and defined by a ~f b if f(a) = f(b). For each of the following functions find f(S) and S/f and exhibit the bijection between them the equivalence classes of S/f for the equivalence relation (a) f : Z → Z12 given by f(n) = [8n]12. (b) f : Z12 → Z12 given by f(a]12) = [5x]12- (c) f : Z24 → Z24 given by f([x]24) = [4x]24.
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Consider the relation {(x, y)|x, y E R, (x – 2)² +y² = 1}. (a) Does the above relation represent a function? Graph the relation and justify your answer. (b) Which of the following symmetries are respected by the above relation? symmetry about x-axis, symmetry about y-axis, symmetry about the origin. Justify your answer.
Let us define a relation S on R3: If p1 = (x1, y1, z1) and p2 = (x2, y2, z2) are two points in R3, then (p1, p2) in S (or p1 ~ p2) if (x1 - x2) + 2(y1 - y2) + 3(z1 - z2) = 0. Describe all equivalence classes. [Continuation of an earlier question.]

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For the function S : Z4s → Za defined as f(z) = 4r + 1 mod 6. Co defined as zRy when f(x) = f(w). Show that this relation is: (a) reflexive; (b) symmetric; (c) transitive; this will prove that R is an equivalence relation; finally (d) calculate its equivalence classes.