Suppose f: A --> B is a function. Define the function F: P(B) --> P(A) (from the powerset of B to the one of A) by setting F(S) = f-1(S) for all S subset of B. Prove or disprove: (1) F is 1-1 if and only if f is onto. (2) F is onto if and only if f is 1-1. [Finite, not infinite.

Suppose f: A --> B is a function. Define the function F: P(B) --> P(A) (from the powerset of B to the one of A) by setting F(S) = f-1(S) for all S subset of B. Prove or disprove: (1) F is 1-1 if and only if f is onto. (2) F is onto if and only if f is 1-1. [Finite, not infinite.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: Let f: A→B be a function. Prove or disprove the equivalence: (a) f is injective (b) If C is any set…

Q: Consider the relation R on the set of functions defined by (f, g) = R if and only if f(x) is…

Q: Give an example of a function that is one-to-one but not onto. Give an example of a function that is…

Q: Give an example of a function from the set of all integers to the set of all negative even integers…

Q: Let A = {1, 2, 3, 4}. For any function f from A to A and any relation R on A, we define a relation S…

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

Q: 3. Let A be a finite set. In this exertise functions f, g will be of type g: A → A and ƒ: A → A. (a)…

Q: Let f : N → P(N) be the function defined by f(n) = {n, n + 1, . . . , 2n} The following statement is…

A: 1) is false and 2) is true.

Q: Determine whether the function f is one-to-one, onto, or both. Prove your answers. The domain and…

Q: ›) Prove that if the function ƒ is 1-1, then ƒ−¹ is a function. Caution! The ONLY assumptions here…

A: We use definition of function and one-one function to prove that f-1 is function.

Q: 1. Let A = {blue, yellow, red}, B = of the relations below (a) determine if it is a function (b) if…

A: R1: D X B={(cat,blue),(cat,red)} So, {(cat,blue)} ⊂{(cat,blue),(cat,red)} Therefore R1 is a…

Q: An ordering function is a function f:Z→Z which takes the set of integers, and maps them to another…

A: Given: A recursive definition to define the set of all possible ordering functions is as follows:…

Q: suppose g: A---> B and f: B---> C are both one-to-one. prove f o g is one-to-one

Q: IF A = { 1,1,2,3,5} B = {2,4} A relation S from A to B such that (x, y) ∈ S means that x ≤ 3 and y…

Q: Give an example of a function from the set of all integers to the set of all positive even integers…

A: Given, Domain = Set of all integers. Co-domain = Set of all positive even integers. The set of all…

Q: 4. Consider the following relations R on the integers Z. For each of these relations, say whether…

A: For the solution of the problem follow the next steps.

Q: (i) Let A = {1,2,3,4,5), then either find a relation on A that is not reflexive on A and not…

A: Solution:

Q: Let Q: A - B be a function and CSA. define the restriction and also the extension of a function

A: Restriction and extension of a function

Q: State if each of the following is a (write the letter only): A. Not a function, just a relation B. A…

Q: Let D be the set of all finite subsets of positive integers. Define a function T:Z* → D as follows:…

A: Since T(n)=The set of positive divisors of n. Since 17 ,5 are prime numbers.

Q: If f is a function having as its domain and range the set N of natural numbers, then f is bijective.

A: For the solution follow the next step.

Q: Let A = {2, 3, 4, 5} and B = {6, 8, 10}. A relation R is defined from A to B by a R b if a | b and…

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

Q: 5. (8 points) Let S4be the set of all strings of 0's and 1's of length 4. Define F:S4 → Znonneg such…

A: The objective of the question is to understand the properties of the function F defined on the set…

Q: Given a relation g from the set D = {1, 2, 3, 4} to the set C = {a, b, c, d} where g= {(1, a), (2,…

A: A relation from the set to the set where .The objective is to determine whether the relation is a…

Q: 13. (9 points) Let D be the set of finite subsets of positive integers. Let S be the set of all…

A: The objective of this question is to find the set of all even factors of given numbers using the…

Q: ) A) Show that the set of all odd integers is countable, by producing a 1-1 unction F From J onto…

Q: Given f: {1,3,5} → {a,b,c,d} defined by the assignment 1b, 3b, 5 → b Is it a function or not? If…

Q: Def (Function): A binary relation RCX X Y is called a function from X to Y, written f:X →Y if -…

A: The concepts have been explained with examples.

Q: State if each of the following is a (write the letter only): A. Not a function, just a relation B. A…

Q: Determine whether the function f from { a, b, c, d} to { a, b, c, d} is injective (one - to - one),…

Q: Suppose f : N → N is an injective function on the natural numbers which, considered as a relation,…

Question

Suppose f: A --> B is a function. Define the function F: P(B) --> P(A) (from the powerset of B to the one of A) by setting F(S) = f^-1(S) for all S subset of B. Prove or disprove:

(1) F is 1-1 if and only if f is onto.

(2) F is onto if and only if f is 1-1.

[Finite, not infinite.]

Expert Solution

This question has been solved!

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

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

Step 2

Step 3

Step 4

Step 5

Trending now

This is a popular solution!

Step by step

Solved in 5 steps with 5 images

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, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

If f:{1,2,3}->{a,b}, f(1)=a f(2)=b f(3)=a then explain why f is onto but not one to one
3. Let A be a finite set. In this exertise functions f, g will be of type g: A →A and f: A → A. (a) For A = {1, 2, 3, 4} define functions f and g, such that f is surjective and g is injective. (b) From (a) write down functions fog and go f, and determine whether they are surjective and/or injective. (c) Independently from the choices of (a), prove that function f is surjective if and only if it is injective.
(a) What is the number of onto functions f : {1,2, 3, 4, 5} → {1, 2, 3}? (b) What is the number of one to one functions f : {1,2, 3} → {1,2, 3, 4, 5}?
If A is set of different triangles in the plane and B is set of all positive real numbers. A relation R is defined from set 4 to set B such that every element of set A is associated with some number in set B which is measure of area of triangle. Is this relation as function?
Given any set of 15 integers, must there be two that have the sameremainder when divided by 12? Prove using the pigeonhole principle. Eitherdescribe the pigeons, the pigeonholes, and how the pigeons get to the pigeonholes, or describe a function by giving its domain, co-domain, and how elements of the domain are related to elements of the co-domain.
please just answer (d)
Let - be a relation on N defined as: a ~b if a < (b – 1). (i) Is ~ reflexive? Prove your answer. (ii) Is ~ (iii) Is ~ transitive? Prove your answer. symmetric? Prove your answer.
Let f be a function from X toY. Define a relation R on X by xRy if f(x) = f(y) Show that R is an equivalence relation. Is there a natural one-to-one function g related to f that maps the set of equivalence classes of R to Y ?
13. (9 points) Let D be the set of finite subsets of positive integers. Let S be the set of all positive integers greate than or equal to 2. Define a function T:S → D as follows: For each integer n ≥ 2, T(n) = the set of all even factors of n. a) Find T(10). b) Find T(17) c) Find T(m), where m is any odd positive integer.
Find two distinct primitive elements of F, over F