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 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 A,B and C be sets, show(Prove) that (A-B)-C =
(A-C)-(B-C).

Let a < b, a, b, ∈ R, and let f : [a, b] → R be continuous
such that f is twice differentiable on (a, b), meaning f is
differentiable on (a, b), and f' is also differentiable on (a, b).
Suppose further that there exists c ∈ (a, b) such that f(a) >
f(c) and f(c) < f(b).
prove that there exists x ∈ (a, b) such that f'(x)=0.
then prove there exists z ∈ (a, b) such...

Let A, B, C and D be sets. Prove that A\B ⊆ C \D if and only if
A ⊆ B ∪C and A∩D ⊆ B

Let F be a field and Aff(F) := {f(x) = ax + b : a, b ∈ F, a ≠ 0}
the affine group of F. Prove that Aff(F) is indeed a group under
function composition. When is Aff(F) abelian?

Let S = {a,b,c,d,e,f,g} and let T = {1,2,3,4,5,6,7,8}.
a. How many diﬀerent functions are there from S to
T?
b. How many diﬀerent one-to-one functions are there from S to
T?
c. How many diﬀerent one-to-one functions are there from T to
S?
d. How many diﬀerent onto functions are there from T to
S?

Let a, b ∈ R. Assume that vector F and vector G are C^1 vector
fields on R^3 . Prove that ∇ × (aF~ + bG~ ) = a∇ × F + b∇ × G.

Let f : A → B and suppose that there exists a function g : B → A
such that (g ◦ f)(a) = a and (f ◦ g)(b) = b. Prove that g = f −1
.
Thank you!

Let f : A → B, and let S, T ⊆ A. Suppose also that f is
one-to-one. Prove that f(S ∩ T) = f(S) ∩ f(T).

Let A and B be sets. Prove that (A∪B)\(A∩B) = (A\B)∪(B\A)

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 10 seconds ago

asked 1 minute ago

asked 1 minute ago

asked 2 minutes ago

asked 3 minutes ago

asked 6 minutes ago

asked 6 minutes ago

asked 7 minutes ago

asked 7 minutes ago

asked 7 minutes ago

asked 7 minutes ago

asked 10 minutes ago