Question

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 stage of the pipeline, f, fails to be onto, then the entire pipeline, h, can also not be onto.

c). Formulate the consequence of theorems 1. and 2. for a bijective h.

d). Prove that if g is not onto, and h is bijective, then f cannot be 1-1.

e). Prove that if f is not 1-1, and h is bijective, then g cannot be onto.

f). Explain what 3-5 implies for our search for finding non-bijective f and g with a bijective composition.

g). Inspired by your finding in 6, find three sets A,B,C, not necessarily different from each other, and two functions f and g so that g is a function from A to B, f is a function from B to C, and h is bijective, while neither f nor g is bijective. Prove that your example is correct. This is the question you need to get completely correct in order to receive any credit for this whole assignment.

h). Give a brief, intuitive explanation of how the specific ways in which your f and g fail to be bijective cancel each other out in a sense and make it possible for h to be bijective.

Answer #1

Let A, B, C be sets and let f : A → B and g : f (A) → C be
one-to-one functions. Prove that their composition g ◦ f , defined
by g ◦ f (x) = g(f (x)), is also one-to-one.

Let g be a function from set A to set B and f be a function from
set B to set C. Assume that f °g is one-to-one and function f is
one-to-one. Using proof by contradiction, prove that function g
must also be one-to-one (in all cases).

Let f : A → B and g : B → C. For each of the statements in this
problem determine if the statement is true or false. No explanation
is required. Just put a T or F to the left of each statement.
a. g ◦ f : A → C
b. If g ◦ f is onto C, then g is onto C.
c. If g ◦ f is 1-1, then g is 1-1.
d. Every subset of...

1.
Let A and B be sets. The set B is of at least the same size as
the set A if and only if (mark all correct answers)
there is a bijection from A to B
there is a one-to-one function from A to B
there is a one-to-one function from B to A
there is an onto function from B to A
A is a proper subset of B
2.
Which of these sets are countable? (mark all...

G is a finite group. We have shown that C(g) ≤ G for any g ∈ G.
Regarding the cosets of C(g):
1. The elements in the same coset all have something in common that
distinguishes them
from the other cosets. Figure out what it is, state it clearly, and
prove it.
2. Find a bijection between cl(g) and the set of cosets G/C(g) =
{ aC(g) | a ∈ G }. State
it clearly and prove that it is...

Let X = [0, 1) and Y = (0, 2).
a. Define a 1-1 function from X to Y that is NOT onto Y . Prove
that it is not onto Y .
b. Define a 1-1 function from Y to X that is NOT onto X. Prove
that it is not onto X.
c. How can we use this to prove that [0, 1) ∼ (0, 2)?

1. Suppose we have the following relation defined on Z. We say
that a ∼ b iff 2 divides a + b. (a) Prove that the relation ∼
defines an equivalence relation on Z. (b) Describe the equivalence
classes under ∼ .
2. Suppose we have the following relation defined on Z. We say
that a ' b iff 3 divides a + b. It is simple to show that that the
relation ' is symmetric, so we will leave...

1. Let A = {1,2,3,4} and let F be the set of all functions f
from A to A. Prove or disprove each of the following
statements.
(a)For all functions f, g, h∈F, if f◦g=f◦h then g=h.
(b)For all functions f, g, h∈F, iff◦g=f◦h and f is one-to-one
then g=h.
(c) For all functions f, g, h ∈ F , if g ◦ f = h ◦ f then g =
h.
(d) For all functions f, g, h ∈...

Product A is made from components B and C. Item B, in turn, is
made from D and E. Item C also is an intermediate item, made from
F and H. Finally, intermediate item E is made from H and G. Note
that item H has two parents. The following are item lead
times:
Item
A
B
C
D
E
F
G
H
Lead Time (weeks)
1
1
2
2
6
4
4
6
a. What lead time (in weeks)...

Let⇀H=〈−y(2 +x), x, yz〉
(a) Show that ⇀∇·⇀H= 0.
(b) Since⇀H is defined and its component functions have
continuous partials on R3, one can prove that there exists a vector
field ⇀F such that ⇀∇×⇀F=⇀H. Show that F =
(1/3xz−1/4y^2z)ˆı+(1/2xyz+2/3yz)ˆ−(1/3x^2+2/3y^2+1/4xy^2)ˆk
satisfies this property.
(c) Let⇀G=〈xz, xyz,−y^2〉. Show that⇀∇×⇀G is also equal to⇀H.
(d) Find a function f such that⇀G=⇀F+⇀∇f.

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