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f d(x, y) denotes the distance between two points

f d(x, y) denotes the distance between two points

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.7: Distinguishable Permutations And Combinations
Problem 30E
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Given the set P of all points in a plane, let M denote the set of all permutations of P that preserve the distance between points, called isometries or motions of the plane. The set of all motions, together with the operation composition o, form a group of isometries (M, o) for the following reasons. If d(x, y) denotes the distance between two points x, y EP, then for any a € M, d(x, y) = d(i(x), (y)) = d(a(a−¹(x)), a(a¯¹(y))) = d(a¯¹(x), a¯¹(y)) since a € M⇒ a ¹(x), a ¹(y) = M. Thefore we conclude that the inverse a-¹ EM too. The identity map EM, and one can also show that a, BEM⇒ Boa € M. O True O False Reset Selection Rationale:
Transcribed Image Text:Given the set P of all points in a plane, let M denote the set of all permutations of P that preserve the distance between points, called isometries or motions of the plane. The set of all motions, together with the operation composition o, form a group of isometries (M, o) for the following reasons. If d(x, y) denotes the distance between two points x, y EP, then for any a € M, d(x, y) = d(i(x), (y)) = d(a(a−¹(x)), a(a¯¹(y))) = d(a¯¹(x), a¯¹(y)) since a € M⇒ a ¹(x), a ¹(y) = M. Thefore we conclude that the inverse a-¹ EM too. The identity map EM, and one can also show that a, BEM⇒ Boa € M. O True O False Reset Selection Rationale:
Given a permutation group (G, o) defined on a set S, let us define a relation ~ on S by a~ b a(x) = y for some a € G, and Vx, y S. Then the relation is an equivalence relation on S, because (i) x ~ x Reflexive (ii) x, y = S with x~y then there is some a € G such that a(x) = y a ¹ (y) = x⇒y~x ⇒ Symmetry -1 : (iii) x, y, z = S with ~ zand z~ y, one can always find a, ß E G such that a(x) = z and B(z) = y. Therefore Boa(x) = B(y)=z⇒x~y⇒ Transitivity. O True O False Reset Selection Rationale:
Transcribed Image Text:Given a permutation group (G, o) defined on a set S, let us define a relation ~ on S by a~ b a(x) = y for some a € G, and Vx, y S. Then the relation is an equivalence relation on S, because (i) x ~ x Reflexive (ii) x, y = S with x~y then there is some a € G such that a(x) = y a ¹ (y) = x⇒y~x ⇒ Symmetry -1 : (iii) x, y, z = S with ~ zand z~ y, one can always find a, ß E G such that a(x) = z and B(z) = y. Therefore Boa(x) = B(y)=z⇒x~y⇒ Transitivity. O True O False Reset Selection Rationale:
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