27. Let A be a set and let f: A → A be a function. Forx, y € A, define x ~y if f (x) = f(y).~ is an equivalence relation on A.%3D(a) Prove that -(b) For A = R and f(x)classes of 0, , and= LJ, find the equivalence%3D734 °

This is a problem of set and function.

Answered: 27. Let A be a set and let f: A A be a…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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7. For each of the following, give an example of functions f: A → B and g: B C that satisfy the stated conditions, or explain why no such example exists. * (a) The function f is a surjection, but the function gof is not a surjection. * (b) The function f is an injection, but the function gof is not an injection. (c) The function g is a surjection, but the function g of is not a surjection. (d) The function g is an injection, but the function go ƒ is not an injection. (e) The function f is not a surjection, but the function gof is a surjection. * (f) The function f is not an injection, but the function gof is an injection. (g) The function g is not a surjection, but the function gof is a surjection. (h) The function g is not an injection, but the function gof is an injection.
Choose the correct option. The number of onto function from the set {1,2,….,11} to the set {1,2,….,10} is- (a)5×⌊11 (b)⌊10 (c)⌊11/2 (d)10×⌊11
Let P(A) be the power set of a non-empty set A A relation R on P(A) is defined as follows: R= {(X,Y): X CY} Example (1) reflexivity, (ii)symmetry and (i) transitivity of R on P(A).
Show that the point of intersection of two lines in the plane of afield F lies in the plane of F.
4) i) Let f: N→ C be a function with f(30) = 0, f(70) = 1 and f(105) = 1. Determine whether f is multiplicative or not. Explain why. Show that f(1) = 0 or f(1) = 1 if f is multiplicative. Determine whether the following functions are multiplicative or not. Give a detailed ii) iii) explanation. a) f(n)= ^ 1 if n 1 mod (2) 2 if n2 mod (2) b) f(n) = 1 if n 1 2 if n2 3 if n 0 mod (3) mod (3) mod (3)
let A = { 1, 2, 3, 4, 5} and B = { a, b, c, d, e, f}. in each case, give an example of a function f : A-->B with indicated properties, or explain why no such function exist. a) f : A--->B, f is one-to-one b) g: B--> A, g is one-to-one c) f : A---> B, f is onto d) g: B-->A, g is onto
17. Let R and S be relations on A= {1, 2, 3, 4} defined by R={(a, b) :b=5-a} and S = {(a, b): a
Do question 4 only
please just answer (d)
On P2, define a relation p(x) ~ q(x) if p(0) = q(0). (a) Show that this is an equivalence relation (b) Find all distinct equivalence classes.

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27. Let A be a set and let f: A A be a function. For x, y E A, define x ~ y if f(x) = f(y). (a) Prove that ~ is an equivalence relation on A'. (b) For A =R and f(x) = [x], find the equivalence classes of 0, , and -.