true or False? Prove your answer! a) Suppose h:R→R is not one-to-one, and suppose a, b∈R with a≠b. Then h(a) =h(b). b) Suppose g:R→R is a function such that for every x∈R, there exists y∈R such that g(x)≠y . Then g is not onto.

true or False? Prove your answer! a) Suppose h:R→R is not one-to-one, and suppose a, b∈R with a≠b. Then h(a) =h(b). b) Suppose g:R→R is a function such that for every x∈R, there exists y∈R such that g(x)≠y . Then g is not onto.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter6: More On Rings

Section6.2: Ring Homomorphisms

Problem 6E

See similar textbooks

Related questions

Q: Test the continuity of the following lons at tne points each (x+1, 3x- 1, 1 < x < 2; (i) x= 1, (ii)…

Q: Exercise 12. Let x, z e F and y, w ɛ F* where F is a field. Prove the following 0, 1, yw'

A: In the question it is asked to prove the following equations given. Bartleby's guidelines: Experts…

Q: Let A = {0, 1} with the discrete topology and let f: A R be defined by f (0) = -1, f(1) = 1.

Q: Let z=x+iy ∈ C where x∈R\{1}, y∈R, and |z|=1. If w = (z+1)/(z-1), then show that Im(w) = y/(x-1).

A: Use the following formulae, to obtain the required result. Real part of a+ib=aImaginary part of…

Q: Let f = 12r + kry + Jryz re-express f O f(r,y,z) = (x, ry, Tyz+) O {r,y) = (x, ry, ryz*) O f(x,y) =…

A: GIVEN f= ix+kxy+jxyz4 ....(1)

Q: \Let X be Hilbert space over field F, and let R, S, T = S(X), 2 > 0: If S≤T then S+R≤T+R. (2) If S≤T…

A: Given : Let X be a Hilbert space over field F Let R,S,T∈SX , λ>0

Q: 2.55 Let P denote the set of all functions f on [0, 1] such that f(x) > 0 for all x € (0, 1]. Is P a…

A: P is not a vector space.

Q: Exercise 2. Let X = C[0, 1], where C[0, 1] is the set of continuous functions on [0, 1], and the…

A: A set (X,d) is called a metric space if these four properties are satisfied for all x, y, z ∈ X: d…

Q: Let A={x e R|-1sx s1} = B, Show that f:A →B given by f (x)=x|x| is a bijecti on.

A: we have to show that given function is bijective

Q: (a) Show (by calculation) that {u1(x), u2(x)} is an orthonormal set. (b) Find the projection p(x) of…

A: Given the space C0,1 of continuous function on the interval 0≤x≤1, with the inner product defined as…

Q: In Exercises 1-7, (X,M) is a measurable space. 1. Let f : X R and Y = f-1(R). Then f is measurable…

A: Given: Let's consider X,M is a measurable space. Here f:X→ℝ¯ and Y=f-1ℝ

Q: Exercise 0.3.23. (a) Prove that the function f(x) : = vex over [0, ∞). 1+ex is strictly con-

A: As per the policy, we are allowed to answer only one question at a time. So, I am answering the…

Q: Q4) Let C(R) be a normed space such that || f -sup{ Ifl,x E R}and T: C (R) -R define Tf)-If I. Is T…

Q: Example (3): suppose X = F", with the inner product defined as (x, y) = E=1*iYt, Vx, y'E X, prove…

A: The objective is to prove that X is Hilbert space.

Q: Suppose V is an inner-product space and u, v e V satisfy ||u|| = ||v|| = 1 and (u, v) = 1. Prove…

Q: Let F(x, y,z)=xi + yj+zk, f(x,y,z)=||F(x,y,z)||. Prove that: Vf" = nf"-²F F v(In f) =

Q: Let X & Y be normed spaces & x # 204. Then BL (X₂Y) is a Banach space in the operator norm if and…

A: Here BL(X,Y) (or, B(X,Y)), is the space of bounded linear operators from X to Y. First ,we show that…

Q: Prove that if r is a vector-valued function that is continuous at c, then || r || is continuous at…

A: Consider the vector r→a=p(a)i+q(a)j+s(a)k. If r is continuous at a=c. So, lima→cr→(t)=r→(c) Then…

Q: consider the relative topologies of the usual topology on R on I₁ = (2,4) and I₂ = (8,16) prove that…

A: Here f is a function from I1 to I2 defined as f(x)= 4x for all x in I1. Where I1=(2,4), I2=(8,16).…

Q: Let f : X → Y be a function where X is a nonempty space and Y is a topological space with a given…

Q: Suppose (X,S,u) is a measure space and f,h: X → F are S-measurable Prove that I|fh||, < ||f||, ||4|a…

A: Given X,S,μ is a measure space and the functions f,h:X→F are S-measurable. Suppose 1p+1q=1r,…

Q: Let X be a normed space, Xn, Yn EX such that Xnx, Yny, then 1-Xn + Yn → 4- ||xn-ynll→→llx - yll.…

A: Given that X is a normed space, xn, yn ∈X such that xn→x, yn→y. To find1. xn+ yn→x+y 2. λxn→λx ∀λ∈F…

Q: Exercise 3. Show that f : Z → N so that f (z) = z² + 1 for all z E Z is neither one-to-one %3D nor…

A: One-One Function: A function is said to be One-One if for every element in the domain there exists…

Q: Show that there exists a set N C (0, 1) that is not Lebesgue measurable.

Q: Exercise 2. Suppose V and W are finite-dimensional, T E L(V, W), and there exists PE W' such that…

A: Given V and W are finite dimensional , T∈LV,W and there exists φ∈W' such that nullT'=spanφ We have…

Q: Let E/F be a finite Galois extension. Let Kị and K2 be intermediate fields

A: Consider the equation

Q: Functional theory and functional analysis

A: Consider the provided question, Let A be an invertible operator acting on space X.

Q: Let r, y EN such that y # 0, then y+r#0.

Q: 3 Let A be the following linear functional on C([0, 1]): xf (Va) da, f e C([0, 1]). Prove that A is…

Q: Prove that the function f(x1,x2) = In (e*i+= + eV=i+1) is convex over R².

Q: Show that if a non-zero element in a normed space X, then there is a bounded linear functional f on…

A: We need to show that if a non-zero element in a normed space X, then there is a bounded linear…

Q: Complete the proof of Theorem 6.2. Theorem 6.2. Let V be an inner product space over F. Then for all…

A: Let V be an inner product space over F. Then for all x ,y € V and c € F,For part (a)It is required…

Q: Let (X, E, µ) be a measure space. Show that f is integrable if and only if fx ]f]dµ < o∞. Moreover…

Q: 9. Let be a bounded open subset of C, and 4: Prove that if there exists a point zo E such that 2 a…

A: please comment if you need any clarification. If you find my answer useful please put thumbs up.…

Q: Let A = Z. Let R be defined on A x A where a Ry if 4 (x- y). Prove that R is transitive.

Q: 2) Let X be normed space and xo + 0 be an element of X. Then using Hahn-Banach Theorem (Normed…

A: We will construct a functional satisfying given data

Q: Let P denote the set of all functions f on [0, 1] such that there exists at least one point x E [0,…

Q: 1. (a) Let f: X → R be a subadditive function defined on a topological vector space. Prove that if f…

A: As you have asked more than one different/unreated to each other questions. We shall solve first…

Q: Let A: C [0,1]→C [0,1]. Prove that operator A is continuous reversible, find A. Ax() = x() + ja-m}xt…

A: Solution:- We have given operator A such that A: C[0,1]- C[0, 1] Also we must know that the inverse…

Q: Prove that Cl(Q) = R in the standard topology on R. %3D

A: The closure of S may equivalently be defined as the union of S and its boundary, and also as the…

Q: LetT be a normal operator on a finite-dimensional inner product space V. Prove that N(T) = N(T*) and…

A: Let T be a normal operator on a finite-dimensional inner product space V

Q: If X is finite dimensional hor med space o T: X-Dy is a Linear operator from X to normed space y.…

Q: Prove that the vector space C of all continuous functions from R to R is infinite-dimensional. Shew…

A: We have to prove given statement:

Q: Suppose U and v are conformally equivalent. Prove that if U is simply connected, then so is V.…

A: Given that U and V are conformally equivalent that is there exist a mapping f from U to V such that…

Q: Let T be a linear operator on a finite-dimensional inner product space V. (a) If T is an orthogonal…

A: Let T be a linear operator on a finite-dimensional inner product space V.(a)Let T be an orthogonal…

Q: IIC sUUset of R". tve R" and let k e R. Prove that S = {x e R":x.v = k} an affine subset of R". %3D

Q: Show the following result: Theorem. Let (X, d) be a metric space, A C X, and let u: A → R be an…

Q: Let X = (!,2) and T3} U = (1,2-)inel,2,U+ jult. %3D n+1 show that X is not T - space

A: This is a problem of Topology.

Q: Use th properties of linear Space A upon t to prove s a) 0.X - o for all X EX b) a.x=0 =>(a=07v…

Q: 1. Let X C R"-1 be bounded. Prove that X considered as a subset of R" is a volume zero set.

A: Given, X⊆ℝn-1 That is, X is a subset of ℝ in terms of exponential n. So, let we consider the set,…

Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

true or False? Prove your answer!

a) Suppose h:R→R is not one-to-one, and suppose a, b∈R with a≠b. Then

h(a) =h(b).

b) Suppose g:R→R is a function such that for every x∈R, there exists y∈R such that g(x)≠y . Then g is not onto.

Expert Solution

Step 1

(a) The given statement, "Suppose $h : R \to R$ is not one-to-one, and suppose $a, b \in R$ with $a \neq b$ then $h (a) = h (b) .$ " is false.

Counter example:- Let $h : R \to R$ defend by $h (x) = x^{2}$

Since function $h (x) = x^{2}$ is not one-one

Take $a = 1$ and $b = 2$

At $a = 1$ , $h (a) = 1$

At $b = 2$ , $h (b) = 4$

Here, $a \neq b$ and $h (a) \neq h (b)$

Hence, statement is false.

Step by step

Solved in 2 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Differential Equation$

Learn more about

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