is defined by , .
Surjection : Let .
Case 1 : If then if we choose then ,
Case 2 : If then if we choose then ,
So for all there exist such that .
Hence is surjective .
One-to-one : Let
So either both n and m are even or both are odd
If both are even then , n= 2k and m=2p say then ,
Also if both m and n are odd the n = 2k+1 and m =2p +1
So if then . So is one-to-one .
As is both one-to-one and surjective so it is an bijection .
.
.
.
.
Please comment if needed .
Is this function a surjection, 1 to 1 or a bijection, or none? Show each property....
23. (a) Show that a function f : X → Y is a surjection if and only if there is a funct io On g : Y → X such that fog = idy. (b) Show that a function : X → Y with nonempty domain X is an injection if and only if there is a function g : Y → X such that g o f-idx. How does this result break down if X = φ? (c) Show...
Suppose that f :X + Y is a surjection and let yo e Y. Define Z = X -f({yo}) (a) Show that the function g: 2 + Y - {yo}, given by g(x) = f(x) for xe Z, well-defined function. (b) Show that g is a surjection. That denotes
Suppose that f:D+Tis a surjection and let to € T. Define Y =Z-F {{to}) CZ. (1) Show that the function g:Y+T - {to}, given by g(t)= f(t) for t e Y, is a well- defined function. (2) Show that g is a surjection.
1. Show that f : (R,Te) → (R,Tj.), given by f(x)-z?, is a continuous bijection whose inverse function is not continuous. Here Tee and Tie are the countable complement and finite complement topologies respectively
Question4 please (1). Let f: Z → Z be given by f(x) = x2. Find F-1(D) where (a) D = {2,4,6,8, 10, 12, 14, 16}. (b) D={-9, -4,0, 16, 25}. (c) D is the set of prime numbers. (d) D = {2k|k Ew} (So D is the set of non-negative integer powers of 2). (2). Suppose that A and B are sets, C is a proper subset of A and F: A + B is a 1-1 function. Show that...
f (A). 7. For each function f : A- B, exhibit an explicit bijection f : Af (a) f:Z-C given by f(k) = it (b) f:Z-Z12 given by f(n) = [8n). (c) f: 212 + Z12 given by f([n]) = [9n]. (a) f: 212 +Z12 given by f([n]) = [5n].
Surjection. Prove F: A => 13 is surjective . YE B. Z = A - F (17.3) CA 1. Show : 9:Z=> 13 - {Yo], given g(x)= fx) for X6 Zis well-defined function 2. Shour: g is surjective
4. Define a function f:N → Z by tof n/2 if n is even 1-(n + 1)/2 if n is odd. f(n) = Show that f is a bijection. 11 ] 7. Let X = R XR and let R be a relation on X defined as follows ((x,y),(w,z)) ER 4 IC ER\ {0} (w = cx and z = cy.) Is R reflexive? Symmetric? Transitive? An equivalence relation? Explain each of your answers. Describe the equivalence classes [(0,0)]R and...
Problem 8. Given each pair of sets, come up with a formula for a bijection between them You do not need to prove your function is a bijection. Your formula should not be complicated by any means 1. From (0, 1) to (211, 2019) 2. From [0, 1) to (0, 1] 3. From NU (o) to N. 4. From the set of even numbers to 2 5. From the set of odd numbers to Z. 6. r2'2 7. From R...
1 Fix an integer N > 1, and consider the function f : [0,1] - R defined as follows: if 2 € (0,1) and there is an integer n with 1 <n<N such that nx € Z, choose n with this property as small as possible, and set f(x) := otherwise set f(x):= 0. Show that f is integrable, and compute Sf. (Hint: a problem from Homework Set 7 may be very useful for 0 this!)