###### 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 ...

(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.) ...

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. ...

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? ...

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 ...

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 ...

(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. ...

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. ...

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 ? ...

Suppose that g(1) = 3 and that  f\circ g=g\...
It suffices to find  g:\mathbb{R}\rightarro...