Question

Let A = {1, 2, 3, 4, 5, 6}. In each of the following, give an example of a function f: A -> A with the indicated properties, or explain why no such function exists.

(a) f is bijective, but is not the identity function f(x) = x.

(b) f is neither one-to-one nor onto.

(c) f is one-to-one, but not onto.

(d) f is onto, but not one-to-one.

Answer #1

1. Let T = {(1, 2), (1, 3), (2, 5), (3, 6), (4, 7)}. T : X ->
Y. X = {1, 2, 3, 4}, Y = {1, 2, 3, 4, 5, 6, 7}
a) Explain why T is or is not a function.
b) What is the domain of T?
c) What is the range of T?
d) Explain why T is or is not one-to one?

Let X = {1, 2, 3, 4} and Y = {2, 3, 4, 5}. Define f : X → Y by
1, 2, 3, 4 → 4, 2, 5, 3. Check that f is one to one and onto and
find the inverse function f -1.

3 Let A = [0, 1) and B = (0, 1). Give an example to a function f
: A → B that is
a) not one to one and not onto
b) onto but not one to one
c) one to one but not onto
d*) one to one and onto

Discrete Math
In this assignment, A, B and C represent sets, g is a function
from A to B, and f is a function from B to C, and h stands for f
composed with g, which goes from A to C.
a). Prove that if the first stage of this pipeline, g, fails to
be 1-1, then the entire pipeline, h can also not be 1-1. You can
prove this directly or contrapositively.
b). Prove that if the second...

Remember that the DOMAIN is INTEGERS
(....,-2,-1,0,1,2,3,4.......) and the TARGET is POSITIVE INTEGERS
(1,2,3,4,.....) .
Give explanation and proofing for each .
Find a function whose domain is the set of all
integers and whose target is the set of all
positive integers that satisfies each set of
properties.
(a) Neither one-to-one, nor onto.
(b) One-to-one, but not onto.
(c) Onto, but not one-to-one.
(d) One-to-one and onto.

Let G = 〈(1 2 3 4 5 6), (1 6)(2 5)(3 4)〉. Let H1 :=
〈(1 4)(2 5)(3 6)〉 and H2 := 〈(1 6)(2 5)(3 4)〉. Determine
if the subgroups H1 and H2 are normal
subgroups of G.

Let
f(x)=ln[x^3(x+5)^5(x^2+4)^6]
f '(x)=

For each problem below, either give an example of a function
satisfying the give conditions, or explain why no such function
exists.
(a) An injective function f:{1,2,3,4,5}→{1,2,3,4}
(b) A surjective function f:{1,2,3,4,5}→{1,2,3,4}
(c) A bijection f:N→E, where E is the set of all positive even
integers
(d) A function f:N→E that is surjective but not injective
(e) A function f:N→E that is injective but not surjective

(6) (a) Give an example of a ring with 9 elements which is not a
field. Explain your
answer.
(b) Give an example of a field of 25 elements. Explain why.
(c) Find a non-zero polynomial f(x) in Z3[x] such that f(a) = 0
for every a ? Z3.
(d) Find the smallest ideal of Q that contains 3/4.

Let A = [1 5 ; 3 1 ; 2 -4] and b = [1 ; 0 ; 3] (where semicolons
represent a new row)
Is equation Ax=b consistent?
Let b(hat) be the orthogonal projection of b onto Col(A). Find
b(hat).
Let x(hat) the least square solution of Ax=b. Use the formula
x(hat) = (A^(T)A)^(−1) A^(T)b to compute x(hat). (A^(T) is A
transpose)
Verify that x(hat) is the solution of Ax=b(hat).

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