Question

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 an uncountable set is uncountable.

e. If f : X → Y and is onto Y , then Y <∼ X.

f. If f −1 and g −1 are functions, then g ◦ f has an inverse.

g. If A is uncountable, then P(A) is uncountable.

h. Every subset of a countable set is countable.

i. Every subset of a denumerable set is denumerable.

Answer #1

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 ∈...

Let f : A → B and g : B → C. For each statement below either
prove it or construct f, g, A, B, C which show that the statement
is false.
(a) If g ◦ f is surjective, then g is surjective.
(b) If g ◦ f is surjective, then f is surjective.
(c) If g ◦ f is injective, then f and g are injective

Let A, B be sets and f: A -> B. For any subsets X,Y subset of
A, X is a subset of Y iff f(x) is a subset of f(Y).
Prove your answer. If the statement is false indicate an
additional hypothesis the would make the statement true.

Let f and g be continuous functions from C to C and let D be a
dense
subset of C, i.e., the closure of D equals to C. Prove that if
f(z) = g(z) for
all x element of D, then f = g on C.

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 Let A = {a, e, g} and B = {c, d, e, f, g}. Let f : A → B and
g : B → A be defined as follows: f = {(a, c), (e, e), (g, d)} g =
{(c, a), (d, e), (e, e), (f, a), (g, g)}
(a) Consider the composed function g ◦ f.
(i) What is the domain of g ◦ f? What is its codomain?
(ii) Find the function g ◦ f. (Find...

Let f and g be functions between A and B. Prove that f = g iff
the domain of f = the domain of g and for every x in the domain of
f, f(x) = g(x).
Thank you!

Let f, g : X −→ C denote continuous functions from the open
subset X of C. Use the properties of limits given in section 16 to
verify the following:
(a) The sum f+g is a continuous function. (b) The product fg is
a continuous function.
(c) The quotient f/g is a continuous function, provided g(z) !=
0 holds for all z ∈ X.

There are two parts of this problem. State whether each
statement is true or false. Please give a clear explanation of
each.
a. Let G be a group and g in G. Define f : G → G by f(a) =
(g-1)(a)(g) or f(a) = g-1ag, for all a in G.Then f is a permutation
on G.
b. Let G/H be the set of left cosets of the subgroup H in G. If
the operation given by the formula aHbH...

Let A be a finite set and let f be a surjection from A to
itself. Show that f is an injection.
Use Theorem 1, 2 and corollary 1.
Theorem 1 : Let B be a finite set and let f be a function on B.
Then f has a right inverse. In other words, there is a function g:
A->B, where A=f[B], such that for each x in A, we have f(g(x)) =
x.
Theorem 2: A right inverse...

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