Prove a (Dedekind) set is infinite iff there exists an injective
function f : N→ A.
please help prove this clearly and i will rate the best answer
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Let f : R → R be a function.
(a) Prove that f is continuous on R if and only if, for every
open set U ⊆ R, the preimage f −1 (U) = {x ∈ R : f(x) ∈ U} is
open.
(b) Use part (a) to prove that if f is continuous on R, its zero
set Z(f) = {x ∈ R : f(x) = 0} is closed.
TOPOLOGY
Let f : X → Y be a function.
Prove that f is one-to-one and onto if and only if f[A^c] =
(f[A])^c for every subset A of X. (prove both directions)
Considering the illustrations for Concave and Convex
mirrors.
Prove using geometry that the reflected rays reach the focal
point f=R/2 in the limit as the incoming rays approach the
principal axis.
Hint: Consider the triangle formed by the radius of curvature,
principal axis, and reflected ray, and use the law of sines.
Proof of If and Only if (IFF) and
Contrapositive
Let x,y be integers. Prove that the product xy is odd if and
only if x and y are both odd integers.
Proof by Contradiction
Use proof by contradiction to show that the difference of any
irrational number and any rational number is irrational. In other
words, prove that if a is irrational and b is a rational numbers,
then a−b is irrational.
Direct Proof
Using a direct proof, prove that:...
Determine the intervals on which the function below is
increasing or decreasing, concave or convex. Determine also
relative maxima and minima, inflections points, symmetry,
asymptotes, and intercepts if any. Then sketch the curve.
. ? = 3?+3/(3? − 3)^2
Some hints: use the definition: f is a function iff a = b
implies f(a) = f(b) and recall that in informal proofs we show an
implication by assuming the if part of the implication, and then
deducing the then part of the implication.
The base case will show that a = b implies f(a) = f(b) when f(x)
= c0 (a constant function). The inductive case will
assume a = b implies f(a) = f(b) for degree k, and...
For Bi-convex lens, Bi-concave lens Concave Mirror and Concave
mirror :
1 ) By looking at the lens and mirrors in person, comment on the
orientation and location of the images you see?
2 )How could these types of lens be used for lighting
purposes?
Prove the following:
Let f and g be real-valued functions defined on (a, infinity).
Suppose that lim{x to infinity} f(x) = L and lim{x to infinity}
g(x) = M, where L and M are real. Then lim{x to infinity} (fg)(x) =
LM.
You must use the following definition: L is the
limit of f, and we write that lim{x to infinity} f(x) = L provided
that for each epsilon > 0 there exists a real number N > a
such...