Mathematical Methods for Physics and Engineering - Matematica.NET

28.8 EXERCISES28.4 Prove that the relationship X ∼ Y , defined by X ∼ Y if Y can be expressed inthe formY = aX + bcX + d ,with a, b, c and d as integers, is an equivalence relation on the set of real numbersR. Identify the class that contains the real number 1.28.5 The following is a ‘proof’ that reflexivity is an unnecessary axiom for an equivalencerelation.Because of symmetry X ∼ Y implies Y ∼ X. Then by transitivity X ∼ Y andY ∼ X imply X ∼ X. Thus symmetry and transitivity imply reflexivity, whichtherefore need not be separately required.Demonstrate the flaw in this proof using the set consisting of all real numbers plusthe number i. Show by investigating the following specific cases that, whether ornot reflexivity actually holds, it cannot be deduced from symmetry and transitivityalone.(a) X ∼ Y if X + Y is real.(b) X ∼ Y if XY is real.28.6 Prove that the set M of matricesA =(a b0 cwhere a, b, c are integers (mod 5) and a ≠0≠ c, form a non-Abelian groupunder matrix multiplication.Show that the subset containing elements of M that are of order 1 or 2 donot form a proper subgroup of M,(a) using Lagrange’s theorem,(b) by direct demonstration that the set is not closed.28.7 S is the set of all 2 × 2 matrices of the form( )w xA =, where wz − xy =1.y zShow that S is a group under matrix multiplication. Which element(s) have order2? Prove that an element A has order 3 if w + z +1=0.28.8 Show that, under matrix multiplication, matrices of the form( )a0 + aM(a 0 , a) =1 i −a 2 + a 3 i,a 2 + a 3 i a 0 − a 1 iwhere a 0 and the components of column matrix a =(a 1 a 2 a 3 ) T are realnumbers satisfying a 2 0 + |a|2 = 1, constitute a group. Deduce that, under thetransformation z → Mz, wherez is any column matrix, |z| 2 is invariant.28.9 If A is a group in which every element other than the identity, I, hasorder2,prove that A is Abelian. Hence show that if X and Y are distinct elements of A,neither being equal to the identity, then the set {I,X,Y ,XY } forms a subgroupof A.Deduce that if B is a group of order 2p, withp a prime greater than 2, then Bmust contain an element of order p.28.10 The group of rotations (excluding reflections and inversions) in three dimensionsthat take a cube into itself is known as the group 432 (or O in the usual chemicalnotation). Show by each of the following methods that this group has 24 elements.1071),