Let X be a set with an equivalence relation ∼. Let f : X/ ∼→ Y...
Let X be a set with an equivalence relation ∼. Let f : X/ ∼→ Y be a function with domain as the quotient set X/ ∼ and codomain as some set Y . We define a function ˜f, called the lift of f, as follows: ˜f : X → Y, x 7→ f([x]). We define a function Φ : F(X/ ∼, Y ) → F(X, Y ), f 7→ ˜f. (1) Is Φ injective? Give a proof or a counterexample. (2) Is Φ surjective? Give a proof or a counterexample.
Homework Answers
![f: XIN Y IS to? pe ] Fixar is lift of sot fop= Rood where þ : x þ : x xa is the equiv. map. [x] (elass of x) tn > Y ie 2 F(x,](http://img.homeworklib.com/questions/18f15940-8741-11eb-8d45-33f8cc7ae079.png?x-oss-process=image/resize,w_560)
2. Let f : A ! B. DeÖne a relation R on A by xRy i§ f (x) = f (y). a. Prove that R is an equivalence relation on A. b. Let Ex = fy 2 A : xRyg be the equivalence class of x 2 A. DeÖne E = fEx : x 2 Ag...
2. Let f : A ! B. DeÖne a relation R on A by xRy i§ f (x) = f
(y). a. Prove that R is an equivalence relation on A. b. Let Ex =
fy 2 A : xRyg be the equivalence class of x 2 A. DeÖne E = fEx : x
2 Ag to be the collection of all equivalence classes. Prove that
the function g : A ! E deÖned by g (x) = Ex is...
Let X = {0, 1, 2} and Y = {0,1,2}. Now we define f={(0,1),(1,0),(2,1)] Please enter...
Let X = {0, 1, 2} and Y = {0,1,2}. Now we define f={(0,1),(1,0),(2,1)] Please enter your answer as a sum of the following numbers (they are not mutually exclusive): • 1 ifff is a function f : X Y • 2 ifff is a function and it is also injective • 4ifff is a function and it is also surjective This means that your answer can be 0 (not a function), 1 (a function but neither injective or surjective)....
Answer the questions in the space provided below. 1. The definition of a function f: X...
Answer the questions in the space provided below. 1. The definition of a function f: X + Y is as a certain subset of the product X x Y. Let f: N + N be the function defined by the equation f(n) = n2. For each pair (x, y) listed below, determine whether or not (x,y) ef. a) (2,4) b) (5, 23) c) (1,1) d) (-3,9) 2. For each function defined below, state whether it is injective (one-to-one) and whether...
Theorem 7.3.5 Let P be a partition of a nonempty set X. Define a relation~on X for all a, b X by ...
Theorem 7.3.5 Let P be a partition of a nonempty set X. Define a relation~on X for all a, b X by defining: Then is an equivalence relation on X. Furthermore, the equivalence classes ofare exactly the elements of the partition P: that is, X/ ~= P. Proof: See page 164 in your textbook. a,b,c,d,e,f partition P = {{a, c, e), {b, f}, {d)) 5 Let A = Give a complete listing of the ordered pairs in the equivalence relation...
Let h : X −→ Y be defined by h(x) := f(x) if...
Let h : X −→ Y be defined by
h(x) :=
f(x) if x ∈ F
g
−1
(x) if x ∈ X − F
Now we must prove that h is injective and bijective. Starting
with injectivity, let x1, x2 ∈
X such that h(x1) = h(x2). Assume x1 ∈ F and x2 ∈ X −F. Then h(x1)
= f(x1) ∈ f(F)
and h(x2) = g
−1
(x2) ∈ g
−1
(X − F) = Y...
2 Functions a. A function f : A-B is called injective or one-to-one if whenever f(x)-f(y) for some x, y E A then x = y. That is Vz, y A f(x) = f(y) → x = y. Which of the following functions are injec...
2 Functions a. A function f : A-B is called injective or one-to-one if whenever f(x)-f(y) for some x, y E A then x = y. That is Vz, y A f(x) = f(y) → x = y. Which of the following functions are injective? In each case explain why or why not i. f:Z-Z given by f() 3r +7 (1 mark ii. f which maps a QUT student number to the last name of the student with that student...
The assertion, as given in the source: (Theorem 3.4.8 of Johnsonbaugh's text.) Let R be an equivalence relation on a set X. For each a in X, define [a] as (xeX | xRaj. Then the following set...
The assertion, as given in the source: (Theorem 3.4.8 of Johnsonbaugh's text.) Let R be an equivalence relation on a set X. For each a in X, define [a] as (xeX | xRaj. Then the following set is a partition of X: S={[a] l a eX). Logical structure of the assertion: Proof framework, based on this logical structure: Previous resulis needed (give one to three previous resulis that are most important for this prooj): Interesting or unexpected tricks, or summary:...
need help with proving discrete math HW, please try write clearly and i will give a thumb up thanks!! Let A and be B be...
need help with proving discrete math HW, please try write
clearly and i will give a thumb up thanks!!
Let A and be B be sets and let f:A B be a function. Define C Ax A by r~y if and only if f(x)f(y). Prove thatis an equivalence relation on A. Let X be the set of~-equivalence classes of A. L.e. Define g : X->B by g(x) Prove that g is a function. Prove that g is injective. Since g...
1. a) Let A = {2n|n ∈ ℤ} (ie, A is the set of even numbers)...
1. a) Let A = {2n|n ∈ ℤ} (ie, A is the set of even numbers) and define function f: ℝ → {0,1}, where f(x) = XA(x) That is, f is the characteristic function of set A; it maps elements of the domain that are in set A (ie, those that are even integers) to 1 and all other elements of the domain to 0. By demonstrating a counter-example, show that the function f is not injective (not one-to-one). b)...
6. Given a finite set A, denote IA] as a nurnber of elements in A. Let f : X → Y be a function wi...
6. Given a finite set A, denote IA] as a nurnber of elements in A. Let f : X → Y be a function with |XI, Yl< oo, i.e. X, Y are finite sets. Prove the following statements a) IXIS IYİ if f is injective. b) IY1S 1X1 if f is surjective.
6. Given a finite set A, denote IA] as a nurnber of elements in A. Let f : X → Y be a function with |XI, Yl
2. Let f : A ! B. DeÖne a relation R on A by xRy i§ f (x) = f
(y). a. Prove that R is an equivalence relation on A. b. Let Ex =
fy 2 A : xRyg be the equivalence class of x 2 A. DeÖne E = fEx : x
2 Ag to be the collection of all equivalence classes. Prove that
the function g : A ! E deÖned by g (x) = Ex is...
Let X = {0, 1, 2} and Y = {0,1,2}. Now we define f={(0,1),(1,0),(2,1)] Please enter your answer as a sum of the following numbers (they are not mutually exclusive): • 1 ifff is a function f : X Y • 2 ifff is a function and it is also injective • 4ifff is a function and it is also surjective This means that your answer can be 0 (not a function), 1 (a function but neither injective or surjective)....
Answer the questions in the space provided below. 1. The definition of a function f: X + Y is as a certain subset of the product X x Y. Let f: N + N be the function defined by the equation f(n) = n2. For each pair (x, y) listed below, determine whether or not (x,y) ef. a) (2,4) b) (5, 23) c) (1,1) d) (-3,9) 2. For each function defined below, state whether it is injective (one-to-one) and whether...
Theorem 7.3.5 Let P be a partition of a nonempty set X. Define a relation~on X for all a, b X by defining: Then is an equivalence relation on X. Furthermore, the equivalence classes ofare exactly the elements of the partition P: that is, X/ ~= P. Proof: See page 164 in your textbook. a,b,c,d,e,f partition P = {{a, c, e), {b, f}, {d)) 5 Let A = Give a complete listing of the ordered pairs in the equivalence relation...
Let h : X −→ Y be defined by
h(x) :=
f(x) if x ∈ F
g
−1
(x) if x ∈ X − F
Now we must prove that h is injective and bijective. Starting
with injectivity, let x1, x2 ∈
X such that h(x1) = h(x2). Assume x1 ∈ F and x2 ∈ X −F. Then h(x1)
= f(x1) ∈ f(F)
and h(x2) = g
−1
(x2) ∈ g
−1
(X − F) = Y...
2 Functions a. A function f : A-B is called injective or one-to-one if whenever f(x)-f(y) for some x, y E A then x = y. That is Vz, y A f(x) = f(y) → x = y. Which of the following functions are injective? In each case explain why or why not i. f:Z-Z given by f() 3r +7 (1 mark ii. f which maps a QUT student number to the last name of the student with that student...
The assertion, as given in the source: (Theorem 3.4.8 of Johnsonbaugh's text.) Let R be an equivalence relation on a set X. For each a in X, define [a] as (xeX | xRaj. Then the following set is a partition of X: S={[a] l a eX). Logical structure of the assertion: Proof framework, based on this logical structure: Previous resulis needed (give one to three previous resulis that are most important for this prooj): Interesting or unexpected tricks, or summary:...
need help with proving discrete math HW, please try write
clearly and i will give a thumb up thanks!!
Let A and be B be sets and let f:A B be a function. Define C Ax A by r~y if and only if f(x)f(y). Prove thatis an equivalence relation on A. Let X be the set of~-equivalence classes of A. L.e. Define g : X->B by g(x) Prove that g is a function. Prove that g is injective. Since g...
6. Given a finite set A, denote IA] as a nurnber of elements in A. Let f : X → Y be a function with |XI, Yl< oo, i.e. X, Y are finite sets. Prove the following statements a) IXIS IYİ if f is injective. b) IY1S 1X1 if f is surjective.
6. Given a finite set A, denote IA] as a nurnber of elements in A. Let f : X → Y be a function with |XI, Yl
Most questions answered within 3 hours.
-
Consider a 3m x 3m window in a house. The thermal conductivity
is reported to be...
asked 13 minutes ago -
Why has California been the favorite destination of large number
of secondary migrants?
asked 46 minutes ago -
Do not neglect the old for the new. The existing business must
not lose priority simply...
asked 3 hours ago -
Kylie is a single mom with two dependent children,
Tanner, age 7 and Olivia, age 11....
asked 5 hours ago -
Phosphorous + bromine = phosphorous tribromide. If 35.0 g of
bromine are reacted and 27.9 grams...
asked 6 hours ago -
Derive the long wavelength limit of the Planck energy density
distribution
asked 6 hours ago -
Calculate the pH of each of the following solutions.
0.50 M HBr
3.1×10−4 M KOH
4.2×10−5...
asked 10 hours ago -
For the year ended December 31, Depot Max’s cost of merchandise
sold was $85,600. Inventory at the...
asked 10 hours ago -
Week 10 - Professional Memo Assignment
Professional Memo Assignment
Your mission for this week, should you...
asked 10 hours ago -
Write a Python program that stores the data for each
player on the team, and it...
asked 10 hours ago -
In
the last 3 months, mike never knows when he is going to get his
allowance...
asked 10 hours ago -
Is Ca(OH)2 a Bronsted base, Lewis base, or both? Why?
asked 10 hours ago