Question

Let X = { x, y, z }. Let the list of open sets of X be Z1. Z1 = { {}, {x}, X }. Let Y = { a, b, c }. Let the list of open sets of Y be Z2. Z2 = { {}, {a, b}, Y }.

Let f : X --> Y be defined as follows: f (x) = a, f (y) = b, f(z) = c

Is f continuous? Prove or disprove using the topological definitions of continuity.

Answer #1

Let X, Y ⊂ Z and x, y ∈ Z Let A = (X\{x}) ∪ {x}.
a) Prove or disprove: A ⊆ X
b) Prove or disprove: X ⊆ A 4
c) Prove or disprove: P(X ∪ Y ) ⊆ P(X) ∪ P(Y ) ∪ P(X ∩ Y )
d) Prove or disprove: P(X) ∪ P(Y ) ∪ P(X ∩ Y ) ⊆ P(X ∪ Y )

Let X, Y ⊂ Z and x, y ∈ Z
Let A = (X\{x}) ∪ {x}.
a) Prove or disprove: A ⊆ X
b) Prove or disprove: X ⊆ A
c) Prove or disprove: P(X ∪ Y ) ⊆ P(X) ∪ P(Y ) ∪ P(X ∩ Y )
d) Prove or disprove: P(X) ∪ P(Y ) ∪ P(X ∩ Y ) ⊆ P(X ∪ Y )

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 F = {A ⊆ Z : |A| < ∞} be the set of all finite sets of
integers. Let R be the relation on F defined by A R B if and only
if |A| = |B|. (a) Prove or disprove: R is reflexive. (b) Prove or
disprove: R is irreflexive. (c) Prove or disprove: R is symmetric.
(d) Prove or disprove: R is antisymmetric. (e) Prove or disprove: R
is transitive. (f) Is R an equivalence relation? Is...

Determine if completeness and transitivity are satisfied for the
following preferences defined on x = (x1, x2) and y = (y1, y2).
(Hints: 1- You have to use z = (z1, z2) to prove or disprove
transitivity. 2- You can disprove by a counter example) — x ≽y iff
x1 > y1 or x1 = y1 and x2 > y2.

Determine if completeness and transitivity are satisfied for the
following preferences defined on x = (x1, x2) and y = (y1, y2).
(Hints: 1- You have to use z = (z1, z2) to prove or disprove
transitivity. 2- You can disprove by a counter example) — x ≽y iff
x1 > y1 or x1 = y1 and x2 > y2.

Determine if completeness and transitivity are satisfied for the
following preferences defined on x = (x1, x2) and y = (y1, y2).
(Hints: 1- You have to use z = (z1, z2) to prove or disprove
transitivity. 2- You can disprove by a counter example) — x ≽y iff
x1 > y1 or x1 = y1 and x2 > y2.

Determine if completeness and transitivity are satisfied for the
following preferences defined on x = (x1, x2)
and y = (y1, y2).
x ≽ y iff x1 > y1 or
x1 = y1 and x2 >
y2.
(Hints: 1- You have to use z = (z1, z2) to
prove or disprove transitivity. 2- You can disprove by a counter
example)

Let x = [1, 1]T , y = [1, 1]T ∈ R 2 and let f : R 2 =⇒ R 2 with
f(z) =z1.x + z2.y for any z = [z1, z2] T ∈ R 2 . Further, z = g(r)
= [r 2 , r3 ] where r ∈ R . Show how chain rule is applied here
giving major steps of the calculation, write down the expression
for ∂f ∂r , and also evaluate ∂f/ ∂r at...

A random variable X has probability density function f(x)
defined by f(x) = cx−6 if x > 1, and f(x) = 0, otherwise.
a. Find the constant c.
b. Calculate E(X) and Var(X).
c. Now assume Z1, Z2, Z3, Z4 are independent RVs whose
distribution is identical to that of X. Compute E[(Z1 +Z2 +Z3
+Z4)/4] and Var[(Z1 +Z2 +Z3 +Z4)/4].
d. Let Y = 1/X, using the formula to find the pdf of Y.

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