Algebra Through Practice: Volume 1, Sets, Relations and Mappings: A Collection of Problems in Algebra with Solutions 1st. Edition by T. S. Blyth, E. F. Robertson | 9780521272858 | 0521272858

Linear Algebra

Algebra Through Practice: Volume 1, Sets, Relations and Mappings: A Collection of Problems in Algebra with Solutions

108 SOLVED PROBLEMS

Question: 3.40

Let N* = {1, 2, 3,…} and for every k ∈ N* define Ik = {X ∈ N* | ½k(k – 1) < x ≤ ½k(k + 1)}. (a) Show that Ik has k elements and that {Ik | k ∈ N} is a partition of N. (b) Define f: N* × N* → N* by f(m, n) = ½{m +n- 2)(m +n-1) + m. Show that f(m, n) ∈ Im + n −1 and deduce that ...

Verified Answer:

(a) It is helpful to compute a few of the

...

Question: 3.41

Let S = R \ { 1 , -1}. Find a mapping f : S → S such that f o f= —idS. (Hint: try mapping ]—1, 1 [ to its complement in S.) ...

Verified Answer:

Consider the mapping

f:S\rightarrow S[/lat...

Question: 3.42

Given mappings f :A → B, g : B → C, h : C → D, suppose that g o f and hog are bijections. Prove that f,g, h are all bijections. ...

Verified Answer:

Since g o f is a bijection we have that g is surje...

Question: 3.43

Let Q+ = {x ∈ Q | x ≥ 0}. If a/b, c/d ∈ Q+ prove that a ⁄ b = c ⁄ d ⇒ |a + b ⁄ hcf(a,b) |= |c + d ⁄ hcf(c,d)|. Deduce that the prescription f(a ⁄ b)=|a + b ⁄ hcf(a, b)| defines a mapping f : Q+ → Q+. Is f a bijection? ...

Verified Answer:

Let

\alpha=\operatorname{hcf}(a,\,b).[/lat...

Question: 3.32

Consider the mapping f : R → R given by f(x) = 4x ⁄ (x² + 1). Sketch the graph of f. Find an interval A = [—k, k] on the x-axis such that (a) {f(x) |x ∈ A} = Im f; (b) g : A → Im f given by g(a) =f(a) for every a ∈ A is a bijection. Obtain a formula for g^-1 : Im f → A ...

Verified Answer:

Use calculus to determine the graph of f (Fig. S3....

Question: 3.44

For mappings f , g : R → R and every λ ∈ R define the mappings f + g, f • g and λ f from R to R in the usual way, namely by setting (f + g)(x)=f(x) +g(x), (f• g)(x)=f(x)g(x), (λf)(x) = λf(x) for every x ∈ R. (a) Show that there are bijections f, g with f + g not a bijection. Show also that there ...

Verified Answer:

(a) Take, for example,

f=\mathrm{id}_{\mat...

Question: 3.33

Sketch the graph of the function f : R → R given by f(x) = 3 + 2x – x². Show that f is not injective. Determine Im f and find a subset A of R such that the restriction of f to A induces a bijection g : A → Im f Obtain a formula for the inverse of this bijection. ...

Verified Answer:

The graph of/is shown in Fig. S3.40. Since, for ex...

Question: 3.34

Let p be a fixed positive integer. Prove that the mapping f : Z → Z given by f(n) = { n+p if n is divisible by p, n if n is not divisible by p, is a bijection, and determine f^−1. ...

Verified Answer:

Note from the definition of f that f(n) is divisib...

Question: 3.37

Let X= {1, 2, 3, 4} and define f: X → X by {f(x) = x + 1 if x ≤ 3, f(4) = 1. Show that there is only one mapping g : X → X with the property that g{1) = 3 and f o g = g o f. Find g. Is it true that there is only one mapping h :X→X with h(1) = 1 and f o h = h o f ? ...

Verified Answer:

Suppose that g(1) = 3 and that

f\circ g=g\...

Question: 3.35

Prove that the mapping f: R → R given by f(x)={x^4 if x ≥ 0; x(2 −x ) if x < 0, is a bijection, and find its inverse. ...

Verified Answer:

It suffices to find

g:\mathbb{R}\rightarro...