Question

Show (prove), from the original definition of the integers, that
subtraction of

integers is well defined.

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Answer #1

Show (prove) that the set of polynomials of degree less than or
equal to 7 with
real coefficients should be uncountable.
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Show (prove) that the set of sequences of 0s and 1s with only
finitely many
nonzero terms should be countable.
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Recall from class that we defined the set of integers by
defining the equivalence relation ∼ on N × N by (a, b) ∼ (c, d) =⇒
a + d = c + b, and then took the integers to be equivalence classes
for this relation, i.e. Z = [(a, b)]∼ | (a, b) ∈ N × N . We then
proceeded to define 0Z = [(0, 0)]∼, 1Z = [(1, 0)]∼, − [(a, b)]∼ =
[(b, a)]∼, [(a, b)]∼...

Recall that we have proven that we have the relation “ mod n” on
the integers where a ≡ b mod n if n | b − a . We call the set of
equivalence classes: Z/nZ. Show that addition and multiplication
are well-defined on the equivalence classes by showing
(a) that you have a definition of addition and multiplication
for pairs which matches your intuition and
(b) that if you choose different representatives when you add or
multiply, the...

Please note n's are superscripted.
(a) Use mathematical induction to prove that 2n+1 +
3n+1 ≤ 2 · 4n for all integers n ≥ 3.
(b) Let f(n) = 2n+1 + 3n+1 and g(n) =
4n. Using the inequality from part (a) prove that f(n) =
O(g(n)). You need to give a rigorous proof derived directly from
the definition of O-notation, without using any theorems from
class. (First, give a complete statement of the definition. Next,
show how f(n) =...

Prove that from any set A which contains 138 distinct integers,
there exists a subset B which contains at least 3 distinct integers
and the sum of the elements in B is divisible by 46. Show all your
steps

Prove that there exist n consecutive positive integers each
having a (nontrivial) square factor. How would you then modify your
proof so that each of these integers instead has a cube factor (or
more generally, a kth power factor where k ≥ 2)?
This is a number theory question.
Please show all steps and make clear notes about what is happening
for a clear understanding.
Please write clearly or do in latex.
Thank you

Consider the function f(x)=arctan((x+5)/(x+4))
What is the slope of the tangent line at the inflection point? The
inflection point I got was: (-9/2, -pi/4).
If you could provide a sketch of the graph as well I'll give a
thumbs up, thank you!

Convert Zr (molar mass = 91.224g/mol) which has a concentration
of 0.026μg/L in seawater to:
0.026μg/L to ppm
0.026μg/L to molality (mol/Kg)
0.026μg/L to molarity (mol/L)
Show work please. Thank you and I will give thumbs up.

can you please show all the steps thank you...
Prove by induction that 3n < 2n for all
n ≥ ______. (You should figure out what number goes in the
blank.)
I know that the answer is n>= 4, nut I need to write the
steps for induction

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