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Chapter 28: Problem 4

Prove that the relationship \(X \sim Y\), defined by \(X \sim Y\) if \(Y\) can be expressed in the form $$ Y=\frac{a X+b}{c X+d} $$ with \(a, b, c\) and \(d\) as integers, is an equivalence relation on the set of real numbers \(\Re\). Identify the class that contains the real number \(1 .\)

Short Answer

Expert verified
Reflexivity, Symmetry, and Transitivity show that X 鈭� Y is an equivalence relation. Class containing 1 is of the form 1 + b.

Step by step solution

01

Verify Reflexivity

To prove reflexivity, demonstrate that for any real number X, the relationship holds that X 鈭� X. Given the definition, set Y = X, then Y = (aX + b) / (cX + d). Choose a=1, b=0, c=0, and d=1: $$X = \frac{1\bullet X + 0}{0 \bullet X + 1} = X$$. The relationship holds, proving reflexivity.
02

Verify Symmetry

To prove symmetry, assume X 鈭� Y, thus Y = (aX + b) / (cX + d). Need to show that Y 鈭� X. Express X in terms of Y:$$ X = \frac{dY - b}{-cY + a}$$, where a, b, c, and d still exist as integers. Therefore, if X 鈭� Y, then Y 鈭� X, proving symmetry.
03

Verify Transitivity

To prove transitivity, assume X 鈭� Y and Y 鈭� Z. Express Y in terms of X: $$Y = \frac{aX + b}{cX + d}$$and Z in terms of Y:$$Z = \frac{eY + f}{gY + h}$$. Substitute Y in Z: $$Z = \frac{e \bullet \frac{aX + b}{cX + d} + f}{g \bullet \frac{aX + b}{cX + d} + h} $$,simplify to get: $$Z = \frac{(ea + fc) X + (eb + fd)}{(ga + hc) X + (gb + hd)}$$, Still of the form: $$Z = \frac{A'X + B'}{C'X + D'}$$ with A', B', C', D' as integers proving transitivity.
04

Identify Class Containing 1

To identify the class containing real number 1, choose form: $$Y = \frac{a \bullet 1 + b}{c \bullet 1 + d} = \frac{a+b}{c+d}$$. Let a=1, b=b, c=0, and d=1, then class containing 1 is of the form: $$Y = \frac{1 + b}{1} = 1 + b$$ for arbitrary integer b.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Reflexivity
Reflexivity is one of the fundamental properties that define an equivalence relation. To say a relation is reflexive means that every element is related to itself. In simpler terms, for every real number X, we must have X 鈭� X to prove reflexivity. Using the relation given in the exercise, that is $$ Y=\frac{aX+b}{cX+d} $$ with integers a, b, c, and d, we can set Y equal to X. By choosing a=1, b=0, c=0, and d=1, we demonstrate that $$ X = \frac{1\bullet X + 0}{0 \bullet X + 1} = X. $$ This satisfies the condition for reflexivity because it shows that any real number X is indeed related to itself.
Symmetry
Symmetry is another crucial property of an equivalence relation. Symmetry means that if a number X is related to another number Y, then Y has to be related to X as well. Given that X 鈭� Y, according to the relation $$ Y=\frac{aX+b}{cX+d}, $$ we need to show that Y 鈭� X. By expressing X in terms of Y, we get $$ X = \frac{dY - b}{-cY + a}. $$ This transformation still fits into the same form, demonstrating that the relation works both ways. Therefore, if X is related to Y (X 鈭� Y), the property of symmetry assures us that Y is also related to X (Y 鈭� X).
Transitivity
The third core concept, transitivity, states that if a number X is related to Y, and Y is related to another number Z, then X must also be related to Z. To prove this, assume X 鈭� Y and Y 鈭� Z. Express Y in terms of X as $$ Y = \frac{aX + b}{cX + d}, $$ and now Z in terms of Y as $$ Z = \frac{eY + f}{gY + h}. $$ Substituting the expression of Y in the relation for Z gives us $$ Z = \frac{e \bullet \frac{aX + b}{cX + d} + f}{g \bullet \frac{aX + b}{cX + d} + h}, $$ which simplifies to $$ Z = \frac{(ea + fc)X + (eb + fd)}{(ga + hc)X + (gb + hd)}. $$ The result of this simplification is still of the same form $$ Z = \frac{A'X + B'}{C'X + D'}, $$ with A', B', C', and D' being integers, thus proving that if X 鈭� Y and Y 鈭� Z, then X 鈭� Z, satisfying the condition of transitivity.

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Most popular questions from this chapter

The group of all non-singular \(n \times n\) matrices is known as the general linear group \(G L(n)\) and that with only real elements as \(G L(n, \mathbf{R}) .\) If \(\mathbf{R}^{*}\) denotes the multiplicative group of non-zero real numbers, prove that the mapping \(\Phi\) : \(G L(n, \mathbf{R}) \rightarrow \mathbf{R}^{*}\), defined by \(\Phi(\mathrm{M})=\operatorname{det} \mathrm{M}\), is a homomorphism. Show that the kernel \(\mathcal{K}\) of \(\Phi\) is a subgroup of \(G L(n, \mathbf{R})\). Determine its cosets and show that they themselves form a group.

The elements of the quaternion group, \(\mathcal{Q}\), are the set $$ \\{1,-1, i,-i, j,-j, k,-k\\} $$ with \(i^{2}=j^{2}=k^{2}=-1, i j=k\) and its cyclic permutations, and \(j i=-k\) and its cyclic permutations. Find the proper subgroups of \(\mathcal{Q}\) and the corresponding cosets. Show that the subgroup of order 2 is a normal subgroup, but that the other subgroups are not. Show that \(\mathcal{Q}\) cannot be isomorphic to the group \(4 \mathrm{~mm}\) \(\left(C_{4 r}\right)\) considered in exercise 28.11.

Consider the following mappings between a permutation group and a cyclic group. (a) Denote by \(A_{n}\) the subset of the permutation group \(S_{n}\) that contains all the even permutations. Show that \(A_{n}\) is a subgroup of \(S_{n}\) (b) List the elements of \(S_{3}\) in cycle notation and identify the subgroup \(A_{3}\). (c) For each element \(X\) of \(S_{3}\), let \(p(X)=1\) if \(X\) belongs to \(A_{3}\) and \(p(X)=-1\) if it does not. Denote by \(\mathcal{C}_{2}\) the multiplicative cyclic group of order \(2 .\) Determine the images of each of the elements of \(S_{3}\) for the following four mappings: $$ \begin{array}{ll} \Phi_{1}: S_{3} \rightarrow \mathcal{C}_{2} & X \rightarrow p(X) \\ \Phi_{2}: S_{3} \rightarrow \mathcal{C}_{2} & X \rightarrow-p(X) \\ \Phi_{3}: S_{3} \rightarrow A_{3} & X \rightarrow X^{2} \\ \Phi_{4}: S_{3} \rightarrow S_{3} & X \rightarrow X^{3} \end{array} $$ (d) For each mapping, determine whether the kernel \(\mathcal{K}\) is a subgroup of \(S_{3}\) and, if so, whether the mapping is a homomorphism.

Given that matrix \(\mathrm{M}\) is a member of the multiplicative group \(G L(3, \mathbf{R})\), determine, for each of the following additional constraints on M (applied separately), whether the subset satisfying the constraint is a subgroup of \(G L(3, \mathbf{R}):\) (a) \(\mathrm{M}^{T}=\mathrm{M}\) (b) \(\mathrm{M}^{T} \mathrm{M}=\mathrm{I}\); (c) \(|\mathrm{M}|=1 ;\) (d) \(M_{i j}=0\) for \(j>i\) and \(M_{i i} \neq 0\).

\(\mathcal{S}\) is the set of all \(2 \times 2\) matrices of the form $$ A=\left(\begin{array}{cc} w & x \\ y & z \end{array}\right), \quad \text { where } w z-x y=1 $$ Show that \(\mathcal{S}\) is a group under matrix multiplication. Which element(s) have order 2 ? Prove that an element \(A\) has order 3 if \(w+z+1=0\).
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