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
If you prove by strong induction a statement of
the form ∀ n ≥ 1P(n), the inductive step proves the following
implications (multiple correct answers are possible):
a) (P(1) ∧ P(2)) => P(3)
b) (P(1) ∧ P(2) ∧ P(3)) => P(4)
c) P(1) => P(2)
a) Prove by induction that if a product of n polynomials is
divisible by an irreducible polynomial p(x) then at least one of
them is divisible by p(x). You can assume without a proof that this
fact is true for two polynomials.
b) Give an example of three polynomials a(x), b(x) and c(x), such
that c(x) divides a(x) ·b(x), but c(x) does not divide neither a(x)
nor b(x).
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.