Answer & Explanation:
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HOMEWORK: Review “Newton’s method for solving equations”
Work an example with x^2 – 2 = 0.
HOMEWORK: Any subset of the counting numbers is either
finite, or the same size as the counting numbers. This makes
the size of the counting numbers, the smallest possible infinite size.
HOMEWORK: Anything I can list has the same size as the
counting numbers. The list does not have to be natural in
any way…it just can’t miss anything. Mathematicians call
listable sets “countable” (usually this excludes finite…
but not always).
SHOW that the rational numbers are listable, hence the set of them is countable.
A complex number that is the root of a polynomial with integer coefficients is called
“algebraic.”
SHOW that the set of algebraic numbers is countable.
SHOW that nonalgebraic numbers exist…not every complex
number equals to root of a polynomial with
integer coefficients.
HOMEWORK: Suppose |S| = m and |T| = n. How many injective
functions are there from S to T? Can you find a formula or
an algorithm for computing this…with justificiation?
How about surjective?
Bijective?
Find injections from P(N) to the reals, and from the reals to P(N).
…
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