What is similarity and difference between first principle
induction mathematical and second principle induction mathematical
?
When to use each principle? are there characteristics that
distinguish the issue to be solved by the first or second
principle?!
Prove using the short north-east diagonals or any other
mathematical method of your preference, that if A is enumerable,
then it is also countable with an enumeration that lists each of
its members exactly three (3) times. Hint. Your proof will consist
of constructing an enumeration with the stated requirement.
In this problem we prove that the Strong Induction Principle and
Induction Principle are essentially equiv-
alent via Well-Ordering Principle.
(a) Assume that (i) there is no positive integer less than 1, (ii)
if n is a positive integer, there is no
positive integer between n and n+1, and (iii) the Principle of
Mathematical Induction is true. Prove
the Well-Ordering Principle: If X is a nonempty set of positive
integers, X contains a least element.
(b) Assume the Well-Ordering Principle...
Draw a convex quadrilateral ABCD, where the diagonals intersect
at point M. Prove: If ABCD is a parallelogram, then M is the
midpoint of each diagonal.
a. Use mathematical induction to prove that for any positive
integer ?, 3 divide ?^3 + 2?
(leaving no remainder).
Hint: you may want to use the formula: (? + ?)^3= ?^3 + 3?^2 * b +
3??^2 + ?^3.
b. Use strong induction to prove that any positive integer ? (? ≥
2) can be written as a
product of primes.
Prove these scenarios by mathematical induction:
(1) Prove n2 < 2n for all integers
n>4
(2) Prove that a finite set with n elements has 2n
subsets
(3) Prove that every amount of postage of 12 cents or more can
be formed using just 4-cent and 5-cent stamps
Use
mathematical induction to prove that for every integer n >=2, if
a set S has n elements, then the number of subsets of S with an
even number of elements equals the number of subsets of S with an
odd number of elements.
pleases send all detail solution.