Question

. Let n ∈ N. Prove (by induction) that n =
2^{kn}m_{n} for some nonnegative kn ∈ Z
and some odd mn ∈ N. (Again, k_{n} and m_{n} may
depend on n.)

Answer #1

3. Let N denote the nonnegative integers, and Z denote the
integers. Define the function g : N→Z defined by g(k) = k/2 for
even k and g(k) = −(k + 1)/2 for odd k. Prove that g is a
bijection.
(a) Prove that g is a function.
(b) Prove that g is an injection
. (c) Prove that g is a surjection.

5. Use strong induction to prove that for every integer n ≥ 6,
we have n = 3a + 4b for some nonnegative integers a and b.

a)
Let R be an equivalence relation defined on some set A. Prove
using induction that R^n is also an equivalence relation. Note: In
order to prove transitivity, you may use the fact that R is
transitive if and only if R^n⊆R for ever positive integer n
b)
Prove or disprove that a partial order cannot have a cycle.

Let R be an equivalence relation defined on some set A.
Prove using mathematical induction that R^n is also an
equivalence relation.

Prove the following: Let n∈Z. Then n2 is odd if and
only if n is odd.

Use induction to prove the following:
- Prove that the sum of the first n odd integers is n2
writing a proof and then a program to go along with it.

Use the Strong Principle of Mathematical Induction to prove that
for each integer n ≥28, there are nonnegative integers x and y such
that n= 5x+ 8y

Let X1, X2, ... be i.i.d. r.v. and N an
independent nonnegative integer valued r.v. Let
SN=X1 +...+ XN.
Assume that the m.g.f. of the Xi, denoted
MX(t), and the m.g.f. of N, denoted MN(t) are
finite in some interval (-δ, δ) around the origin.
1. Express the m.g.f. MS_N(t) of SN in terms
ofMX(t) and MN(t).
2. Give an alternate proof of Wald's identity by computing the
expectation E[SN] as M'S_N(0).
3. Express the second moment E[SN2] in terms...

Prove by induction that if n is an odd natural number,
then 7n+1 is divisible by 8.

1) Prove by induction that 1-1/2 + 1/3 -1/4 + ... - (-1)^n /n is always
positive
2) Prove by induction that for all positive integers n, (n^2+n+1)
is odd.

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