Let A = {1, 2, 3, 4, 5, 6}. In each of the following, give an...
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.
Let Let A = {a, e, g} and B = {c, d, e, f, g}. Let...
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 X = [0, 1) and Y = (0, 2).
a. Define a 1-1 function from...
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)?
Let B = {0, 1}3 = {(0, 0, 0), . . . ,(1, 1, 1)} and...
Let B = {0, 1}3 = {(0, 0, 0), . . . ,(1, 1, 1)} and F
= {Bi : i ∈ I} be the indexed family of sets where
I = {0, 1, 2, 3};
Bi = {(b1, b2, b3) ∈
B : b1 + b2 + b3 = i}.
Calculate the elements of F and show that F is a partition of
B
B.) Let R be the region between the curves y = x^3 , y = 0,...
B.) Let R be the region between the curves y = x^3 , y = 0, x =
1, x = 2. Use the method of cylindrical shells to compute the
volume of the solid obtained by rotating R about the y-axis.
C.) The curve x(t) = sin (π t) y(t) = t^2 − t has two tangent
lines at the point (0, 0). List both of them. Give your answer in
the form y = mx + b ?...
Let a, b, c, d be real numbers with a < b and c < d....
Let a, b, c, d be real numbers with a < b and c < d.
(a) Show that there is a one to one and onto function from the
interval (a, b) to the interval (c, d).
(b) Show that there is a one to one and onto function from the
interval (a, b] to the interval (c, d].
(c) Show that there is a one to one and onto function from the
interval (a, b) to R.
1. a) Let f : C → D be a function. Prove that if C1
and...
1. a) Let f : C → D be a function. Prove that if C1
and C2 be two subsets of C, then
f(C1ꓴC2) = f(C1) ꓴ
f(C2).
b) Let f : C → D be a function. Let C1 and C2
be subsets of C. Give an example of
sets C, C1, C2 and D for which f(C ꓵ D) ≠
f(C1) ꓵ f(C2).
Let X = {1, 2, 3, 4} and Y = {2, 3, 4, 5}. Define f...
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.