Question

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.

Answer #1

Let X, Y and Z be sets. Let f : X → Y and g : Y → Z functions.
(a) (3 Pts.) Show that if g ◦ f is an injective function, then f is
an injective function. (b) (2 Pts.) Find examples of sets X, Y and
Z and functions f : X → Y and g : Y → Z such that g ◦ f is
injective but g is not injective. (c) (3 Pts.) Show that...

Let A, B ⊆R be intervals. Let f: A →R and g: B →R be
diﬀerentiable and such that f(A) ⊆ B. Recall that, by the Chain
Rule, the composition g◦f: A →R is diﬀerentiable as well, and the
formula
(g◦f)'(x) = g'(f(x))f'(x)
holds for all x ∈ A. Assume now that both f and g are twice
diﬀerentiable.
(a) Prove that the composition g ◦ f is twice diﬀerentiable as
well, and ﬁnd a formula for the second derivative...

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!

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...

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,BA,B, and CC be sets such that |A|=11|A|=11, |B|=7|B|=7
and |C|=10|C|=10. For each element (x,y)∈A×A(x,y)∈A×A, we associate
with it a one-to-one function f(x,y):B→Cf(x,y):B→C. Prove that
there will be two distinct elements of A×AA×A whose associated
functions have the same range.

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...

8.4: Let f : X → Y and g : Y→ Z be maps. Prove that if
composition g o f is surjective then g is surjective.
8.5: Let f : X → Y and g : Y→ Z be bijections. Prove that if
composition g o f is bijective then f is bijective.
8.6: Let f : X → Y and g : Y→ Z be maps. Prove that if
composition g o f is bijective then f is...

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...

Proof: Let f and g be functions defined on (possibly different)
closed intervals, and assume the range of f is contained in the
domain of g so that the composition g ◦ f is properly defined If f
is integrable and g is increasing, then g ◦ f is integrable.

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