Answered: 7. Find a counterexample or a proof for…

7. Find a counterexample or a proof for the following statement: If f: A- B and g : B- C are functions such that g • f: A- Cis onto, then fis 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 10E: For each of the following parts, give an example of a mapping from E to E that satisfies the given...

See similar textbooks

Related questions

Q: Determine whether or not the function f : Z × (Z − {0}) → Z is onto, if f((m, n)) = ⌊mn ⌋.

Q: Prove that the function f : R → R, given by f(x) = 2x, is one-one and onto.

A: Given, function f : R → R, given by f(x) = 2x.

Q: 1. Let f : Z6 → Z6 be such that f(7) = x2 + 3. (a) Is f a well defined function? (b) Is f…

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

Q: Determine if the function f : RxR → R, defined by f(x, y) = x2", for all (x, y) E R×R, is…

A: f is not one-one But, f is onto.

Q: Let f : R → R be f(r) = + 1+ \|° (a) Is ƒ everywhere defined? If not, give the domaim (b) Is ƒ onto?…

A: Given, f: R→R be f(x)=x1+|x|

Q: Show that the function f : R → R,defined as f(x) = x2 , is neither one-one nor onto.

A: We will show this by counter examples.Let x= 2 and -2 Then f(2)= 2^2 = 4 and f(-2)= (-2)^2 = 4 This…

Q: Are the following functions 1-1, onto, both, or neither. f: R-R. f(x) = x13 h: Z- Z. h(x) = x g:…

Q: Q6) Prove only one of the following: 1- Let X, Y and Z be nonempty sets. Let f:X -Y, g:Y Z are…

A: Introduction - Injective function - An injective function is a function f that maps distinct…

Q: Let ƒ : A → B be a function. Prove that f is onto if and only if there exists a function g : B → A…

Q: The function f : R → R is defined by the rule f(x)=2x+3 a. Prove f(x) is one to one…

Q: 2. Consider the following four functions from R into R, determine if onto or 1-1 function. Support…

A: To find out which of the following function is one-one, onto .

Q: 2. Determine whether each of these functions from {a,b,c,d} to itself is one-to-one. a. f(a)=b,…

Q: Consider a relation R on Z defined as follows: xRy → xy is even. a. Give a counterexample to show…

Q: For an arbitrary function f: XY, prove that the following relations hold: a) f(Uier Ai) = Uiel…

Q: 1. Let f : Z6 → Z6 be such that f(a) = x2 + 3. (a) Is f a well defined function? (b) Is f…

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

Q: Suppose h is a function such that h(3) = -2, h'(3) = 2, h"(3) = 4, h(5) = 6, h'(5) = 7, h"(5) = 13,…

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

Q: Prove directly from the definition that if X and Y are independent and f and g are measurable…

A: Solution: Given X and Y are independent and f and g are measurable function then we have to prove…

Q: 5. Determine whether the function f : Z × Z →Z is onto if a) f (m, n) = m² + n² b)f (m, n) = m³

A: See the solution.

Q: Exercise 2.12. Determine whether or not the given functions f:Z Z are one-to-one or onto (or neither…

A: The solution of given functions is given as follows :

Q: 1. Consider the functions f : A → B and g : B → C. (a) If f and g are both surjective, show that so…

Q: For a bijective function f:(X,r1)→ (Y,12), one of the following statements is true: O f is open if…

Q: (a) Prove that there exists a function f : Z → Z so that f is one-to-one but f is not onto. (b)…

Q: Is the function f(x) = x + 1 from the set of integers to the set of integers onto?

A: For onto from set A to B, Every value of set B should exist atleast once in the function.

Q: Prove that for functions f : R→R, the e- 6 definition of continuity implies the open set definition.

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: (a) Prove the following relation: В (x + 1,у): B(x, y) x +y %3D

A: We have give Beta function relation, we have to show left hand side is equal to right hand side.

Q: Show that the function Arg(z) is not continuous on each point of the non-positive real axis.

Q: Let f : A → B and g : B C be surjective functions. Prove or disprove whether gof is surjective.

A: Here given that f :A→B and g :B→C be both surjective function . Now we…

Q: Consider a function f:C C, defined by f(z) = 2z2 Then which of the following is true? Check all that…

A: Injective ( one to one): If x1 not equal to x2 then fx1 not equal to fx2. If x1 = x2 then fx1 = fx2.…

Q: (Ь) Give explicit, concrete definitions for two functions f1, f2 : Z → z+ such that: i. f2 is onto…

Q: 4. Give an example of a function from N to N that is (a) one-to-one but not onto. (b) onto but not…

Q: (c) Suppose n = 5 and m = 4, determine the number of onto functions from A to B.

Q: Suppose that f : A → B is a function, C C A and DC B. (a) If f is surjective, prove that D = =…

A: Given- Suppose that f : A → B is a function, C ⊆ A and D ⊆ B. To Prove- If f is surjective, prove…

Q: Let f : A → B be a function. Prove that f is surjective if and only if, for every pair of functions…

A: Please comment if you need any clarification. IF you find my answer useful put thumbs up. Thank you.

Q: (a) Prove that there exists a function f: Z → Z so that f is one-to-one but f is not onto. (b) Prove…

Q: Is the function f(x) = x2 from the set of integers to the set of integers onto?

Q: Which of the following function f: Z XZ → Z is not onto? f(a, b) = a O f(a, b) = |b| f(a, b) = a - b…

Q: On one side are the two parts of the definition of a function f: X -> Y, the definition of…

A: Definition : A function is an equation for which any x that can be plugged into the equation will…

Q: Prove or provide a counterexample to the following statement: if f : A → B and g : B → C are…

A: i have used the definition of onto functions and showed that c has no preimage under f but gof is…

Q: ) Let f: A-B be a function which is not surjective and g : B C which is bijective. Prove that g of:…

A: See full proof below

Q: or give a counterexample to the following stateme ons, then ƒ Ug is a function from DU B to C UA et…

A: Function is also a relation

Q: Let Z4 denote the set of positive integers and consider the function f : Z+ → Z, defined by f(n) =…

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: (a) Prove that there exists a function f : Z→ Z so that f is one-to-one but f is not onto. (b) Prove…

Q: Let f : A B be a function. Prove that f is surjective if and only if, for every pair of functions g…

Q: Which of the following function f: R X R → R is not onto? 1 2x - 3 2 3x3 - x 3 x2-2x…

A: The first function is fx=2x-3 which is linear and hence onto The second function is fx=3x3-x solve…

Q: Let g be defined on an interval A, and let c ∈ A. (a) Explain why g'(c) in Definition…

A: Given that function g is gefined on interval A and c∈A

Q: Show that the following statement is false by drawing a graph that provides a counterexample: If f…

Question

7. Find a counterexample or a proof for the following statement: If f: A - Band g: B-
C are functions such that g • f:A- Cis onto, then fis 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 with 2 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Power Series$

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.

Similar questions

For each of the following parts, give an example of a mapping from E to E that satisfies the given conditions. a. one-to-one and onto b. one-to-one and not onto c. onto and not one-to-one d. not one-to-one and not onto
18. Let and be defined as follows. In each case, compute for arbitrary . a. b. c. d. e.
For the given f:ZZ, decide whether f is onto and whether it is one-to-one. Prove that your decisions are correct. a. f(x)={ x2ifxiseven0ifxisodd b. f(x)={ 0ifxiseven2xifxisodd c. f(x)={ 2x+1ifxisevenx+12ifxisodd d. f(x)={ x2ifxisevenx32ifxisodd e. f(x)={ 3xifxiseven2xifxisodd f. f(x)={ 2x1ifxiseven2xifxisodd
27. Let , where and are nonempty. Prove that has the property that for every subset of if and only if is one-to-one. (Compare with Exercise 15 b.). 15. b. For the mapping , show that if , then .
Let a and b be constant integers with a0, and let the mapping f:ZZ be defined by f(x)=ax+b. Prove that f is one-to-one. Prove that f is onto if and only if a=1 or a=1.
Label each of the following statements as either true or false. 3. Let where A and B are nonempty. Then for every subset S of A.
(See exercise 24.) Show that the relation f(x)Rg(x) if and only if f(x)=g(x) is an equivalence relation on R[ x ]. Describe the equivalence class [ f(x) ]. For each f(x)=i=0naixi in R[ x ], the formal derivative of f(x) is the polynomial f(x)=i=1niaixi1. (For n=0, f(x)=0 by definition.) Prove that [ f(x)+g(x) ]=f(x)+g(x). Prove that [ f(x)g(x) ]=f(x)g(x)+f(x)g(x).