Question

Prove or disprove the following statement: 2^(n+k) is an element of O(2^n) for all constant integer values of k>0.

Answer #1

(a) Prove or disprove the statement (where n is an integer): If
3n + 2 is even, then n is even.
(b) Prove or disprove the statement: For irrational numbers x
and y, the product xy is irrational.

Prove or disprove the following statements. Remember to disprove
a statement you have to show that the statement is false.
Equivalently, you can prove that the negation of the statement is
true. Clearly state it, if a statement is True or False. In your
proof, you can use ”obvious facts” and simple theorems that we have
proved previously in lecture.
(a) For all real numbers x and y, “if x and y are irrational,
then x+y is irrational”.
(b) For...

Prove or disprove that 3|(n^3 − n) for every positive integer
n.

Part #1:
Prove or disprove (formally or informally): The sum of an
integer and its cube is even.
Part #2:
Provide counterexamples to the following statements.
If n2 > 0 then n > 0.
If n is an even number, then n2 + 1 is prime.
(n2 is n to the power of 2).

Prove that if for epsilon >0 there exists a positive integer n
such that for all n>N we have p_n is an element of
(x+(-epsilon),x+epsilon) then p_1,p_2, ... p_n converges to
x.

Prove the following theorem: For every integer n, there is an
even integer k such that
n ≤ k+1 < n + 2.
Your proof must be succinct and cannot contain more than 60
words, with equations or inequalities counting as one word. Type
your proof into the answer box. If you need to use the less than or
equal symbol, you can type it as <= or ≤, but the proof can be
completed without it.

Prove the following theorem: For every integer n, there is an even
integer k such that
n ≤ k+1 < n + 2.
Your proof must be succinct and cannot contain
more than 60 words, with equations or inequalities
counting as one word. Type your proof into the answer box. If you
need to use the less than or equal symbol, you can type it as <=
or ≤, but the proof can be completed without it.

Prove or disprove the following statements.
a) ∀a, b ∈ N, if ∃x, y ∈ Z and ∃k ∈ N such that ax + by = k,
then gcd(a, b) = k
b) ∀a, b ∈ Z, if 3 | (a 2 + b 2 ), then 3 | a and 3 | b.

Let a be an element of order n in a group and d = gcd(n,k) where
k is a positive integer.
a) Prove that <a^k> = <a^d>
b) Prove that |a^k| = n/d
c) Use the parts you proved above to find all the cyclic
subgroups and their orders when |a| = 100.

Consider the following statement: if n is an integer,
then 3 divides n3 + 2n.
(a) Prove the statement using cases.
(b) Prove the statement for all n ≥ 0 using
induction.

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 1 minute ago

asked 21 minutes ago

asked 27 minutes ago

asked 36 minutes ago

asked 42 minutes ago

asked 45 minutes ago

asked 45 minutes ago

asked 45 minutes ago

asked 54 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago