Answered: Let a and b be constant integers with a…

Let a and b be constant integers with a * 0, and let the mappingf: Z ----Z be defined by f(x) = ax + b. Prove thatf is one-to-one

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.2: Mappings

Problem 23E: Let a and b be constant integers with a0, and let the mapping f:ZZ be defined by f(x)=ax+b. Prove...

See similar textbooks

Related questions

Q: If G is simply connected and f : G → ℂ is analytic in G, then f has a primitive in G f has no…

A: It is given that, G is simply connected and f : G→ℂ is an analytic in G. Then, we have to choose…

Q: Consider the following relation < over reals: x < y iff (x– y) e Z. Prove that it is an equivalence…

Q: Define f : Zmn → Zm × Zn by ƒ ([x]mn) = ([x]m, [x]n). Show that f is a function and that f is onto…

A: The function f:ℤmn→ℤm×ℤn is defined by fxmn=xm,xn Show f is a function and f is…

Q: Let w be a primitive 12th root of unity over Q. Find the minimal polynomial for wª over Q.

Q: 2. Let a, b e R and let f be the function Z + Z defined by f(z) - ar + b. For which values of a and…

Q: let G={2nEIJ.] show that (I, +) = (G₁.)

Q: Let E be the algebraic closure of F. Show that every polynomial in F[x] splits in E.

Q: Prove that there is an additive function T : R --> R that is not linear.

A: Let T: V→W be the linear transformation and we it is linear,…

Q: Prove that the map f : Zn → Zm defined by f([x]n) = [x]m is a well-defined function if and only if m…

A: To Prove- Prove that the map f : Zn → Zm defined by fxn = xm is a well-defined function if and only…

Q: Suppose both f(z) and the conjugate of f(z) are holomorphic on a connected open set. Show that f is…

Q: Give an example of onto but not 1-1 function f: 7⁰ → Z>⁰. tit 11-C.

Q: Let D be an integral domain and define a relation ~ on DxD such that b is not equal to 0 and d is…

Q: Let Z be the set of integers and f:Z Z be defined as f(x)=-2x-3. Then the inverse function of exists…

Q: Prove that the residue of the following functions at infinity are -1 and 1 respectively. 2 z³ -z² +1…

Q: Let R be the relation on N defined by x Ry if x and y share a common factor other than 1. Determine…

A: The relation R is defined by xRy if x and y share a common factor. We need to check reflexivity and…

Q: Determine whether the relation R on the set of all integers is reflexive, symmetric, antisymmetric,…

A: Disclaimer: Since you have posted a question with multiple sub-parts, we will solve the first three…

Q: If E is an extension of F and f(x) belong to F[x] and if phi is an automorphism of E leaving every…

Q: . Let f (x) = 11x +9x +21. Prove that f (x) is irreducible over Q.

A: We solve this problem using (mod p ) irreducibility test over f(x) where p = 2 , a prime number.

Q: Let alpha be a zero of f(x) = x2 +2x + 2 in some extension field of Z3. Find the other zero of f(x)…

A: Given: f(x) = x2 +2x + 2

Q: Let D be an integral domain and define a relation - on DxD such that b 0 and d=0 by (a,b) - (c,d) if…

A: Equivalence Relation Definition : A relation R on a set A is said to be an equivalence relation if…

Q: Let L be a linear operator on R1 and let a = L (1). Show that L (x) = ax for all x ∈ R1.

A: It's a highly simple question.

Q: Show that the function f : N→ N, given by f (1) = f(2) = 1 and f(x) = x – 1,for every x > 2, is…

A: Given, the function f : N→ N, given by f (1) = f(2) = 1 and f(x) = x – 1, for every x > 2.

Q: Let p be an integral domain and define a relation - on DxD such that bz0 and dz0 by (a,b) -~(c,d) if…

A: R is the symbol for relation. Symmetric: if (a,b)R(c,d) ∈ D×D ⇒ad=bc ⇒bc=ad ⇒(c,d) R…

Q: Show that the function f: Real #s to Real #s defined by f(x)=x2 is not a homomorphism.

Q: Use a purely group theoretic argument to show that if F is a fieldof order pn, then every element of…

Q: If E is an extension of F and f(x) belong to F(x) and if phi is an automorphism of E leaving every…

Q: Let f,g,h: A→ A be any bijective functions. Then (fogoh)-1 f-1og1oh-1 h-1og-1of-1 O is equal to the…

Q: Let Qc(x) = x2 + c. Prove that if c 1 such that Qc is topologically conjugate to Fµ(x) = µx(1 − x)…

Q: Prove that is G1 is isomorphic to G2 and H1 is isomorphic to H2, then G1 x H1 is isomorphic to G2 x…

Q: Prove or disprove the relation C on Z defined by C = {(x, y): x=y (mod 5)} satisfies the two…

A: I just used the definition of congruence and divisibility

Q: Suppose that f(x) and g(x) are irreducible over F and that deg f(x) and deg g(x) are relatively…

Q: . Let a := V1+ v3. Find the minimal polynomial of a over Q.

Q: Let [a] be a non-zero element of Z43. w| 42. When is an element of Z43 said to be a primitive root?

A: This question is based on primitive roots. We recall the definition and give couple of examples.

Q: Let C = {a, b, c, d}. Find a relation R on C that has exactly 3 ordered pair members and is both…

Q: A relation < is defined on R by x < y if and only if there exists k E N such that x = y + 3k. Prove…

A: Partial order relation

Q: Determine whether the relation R on the set of all integers is reflexive, symmetric, antisymmetric,…

Q: .Let Z2 be an algebraic closure of Z2, and let a, ß e Z2 be zeros of x3 x2+1 and of x3 +x + 1,…

A: We know that, a function h(x) (say, of degree 3) is reducible over ℤ2, if and only if it has zero in…

Q: . Let R be the relation on N defined by xR y if x and y share a common factor other than 1.…

Q: Prove that the mapping x --> x6 from C* to C* is a homomorphism.What is the kernel?

A: Given: the mapping x --> x6 from C* to C* is a homomorphism.

Q: Let A be a nonempty set and let f : A→ A be a function. Prove that if f o f = i4, then f is…

Q: Let R be a relation on the set of all integers such that aRb if and only if a + b is even. Is R…

Q: 6. (a) Let ~ be the relation on Mat2(R) defined by: A ~ B if and only if the (2, 1) entry of A – B"…

A: This problem can be solved as follows:

Q: Determine whether the function f: Z x Z -→ Z is onto if f (m, n) = .m + n

A: We will solve the problem using the definition of onto function.

Q: Define a relation R on Z by declaring that xR y if and only if x^2 ≡ y^2 (mod 4). Prove that R is…

Q: On the set {(a, b)} of all ordered pairs of positive integers, define (x1, Y1) ~ (x2, Y2) if and…

Q: If E is an extension of F and f(x) belong to F[x] and if phi is an automorphism of E leaving every…

Q: Given a closed set A, construct a continuously differentiable func- tion that has A as its set of…

Q: Let ∼ be the relation on Mat2(R) defined by: A ∼ B if and only if the (2, 1) entry of A − BT equals…

Question

Let a and b be constant integers with a * 0, and let the mappingf: Z ----Z be defined by f(x) = ax + b.

Prove thatf is one-to-one

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Step by step

Solved in 2 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Groups$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.

Similar questions

A relation R on a nonempty set A is called asymmetric if, for x and y in A, xRy implies yRx. Which of the relations in Exercise 2 areasymmetric? In each of the following parts, a relation R is defined on the set of all integers. Determine in each case whether or not R is reflexive, symmetric, or transitive. Justify your answers. a. xRy if and only if x=2y. b. xRy if and only if x=y. c. xRy if and only if y=xk for some k in . d. xRy if and only if xy. e. xRy if and only if xy. f. xRy if and only if x=|y|. g. xRy if and only if |x||y+1|. h. xRy if and only if xy i. xRy if and only if xy j. xRy if and only if |xy|=1. k. xRy if and only if |xy|1.
Let f:AA, where A is nonempty. Prove that f a has right inverse if and only if f(f1(T))=T for every subset T of A.
4. Let , where is nonempty. Prove that a has left inverse if and only if for every subset of .
3. Let be an integral domain with positive characteristic. Prove that all nonzero elements of have the same additive order .
Prove that if f is a permutation on A, then (f1)1=f.
Prove that if a subring R of an integral domain D contains the unity element of D, then R is an integral domain. [Type here][Type here]
23. Let be the equivalence relation on defined by if and only if there exists an element in such that .If , find , the equivalence class containing.
Let f(x),g(x),h(x)F[x] where f(x) and g(x) are relatively prime. If h(x)f(x), prove that h(x) and g(x) are relatively prime.
21. A relation on a nonempty set is called irreflexive if for all. Which of the relations in Exercise 2 are irreflexive? 2. In each of the following parts, a relation is defined on the set of all integers. Determine in each case whether or not is reflexive, symmetric, or transitive. Justify your answers. a. if and only if b. if and only if c. if and only if for some in . d. if and only if e. if and only if f. if and only if g. if and only if h. if and only if i. if and only if j. if and only if. k. if and only if.
For an element x of an ordered integral domain D, the absolute value | x | is defined by | x |={ xifx0xif0x Prove that | x |=| x | for all xD. Prove that | x |x| x | for all xD. Prove that | xy |=| x || y | for all x,yD. Prove that | x+y || x |+| y | for all x,yD. Prove that | | x || y | || xy | for all x,yD.
Complete the proof of Theorem 5.30 by providing the following statements, where and are arbitrary elements of and ordered integral domain. If and, then. One and only one of the following statements is true: . Theorem 5.30 Properties of Suppose that is an ordered integral domain. The relation has the following properties, whereand are arbitrary elements of. If then. If and then. If and then. One and only one of the following statements is true: .