Question

1. For each statement that is true, give a proof and for each false statement, give a counterexample

(a) For all natural numbers n, n^{2}
+n + 17 is prime.

(b) p Þ q and ~ p Þ ~ q are NOT logically equivalent.

(c) For every real number *x*
*³* *1,* *x**2**£*
*x**3*.

(d) No rational number x satisfies
*x^**4**+* *1/**x*
*-(**x+1)^(1/2)**=0.*

(e) There do not exist irrational numbers
*x and y such that* *log**x* *y is
rational*

(f) *n^**2**+3n+7* is
odd for all integers n.* *

Answer #1

1. Give a direct proof that the product of two odd integers is
odd.
2. Give an indirect proof that if 2n 3 + 3n + 4 is odd, then n
is odd.
3. Give a proof by contradiction that if 2n 3 + 3n + 4 is odd,
then n is odd. Hint: Your proofs for problems 2 and 3 should be
different even though your proving the same theorem.
4. Give a counter example to the proposition: Every...

Ex 2. Prove by contradiction the following claims. In each proof
highlight what is the contradiction (i.e. identify the proposition
Q such that you have Q ∧ (∼Q)).
Claim 1: The sum of a rational number and an irrational number
is irrational. (Recall that x is said to be a rational number if
there exist integers a and b, with b 6= 0 such that x = a b ).
Claim 2: There is no smallest rational number strictly greater...

You’re the grader. To each “Proof”, assign one of the following
grades:
• A (correct), if the claim and proof are correct, even if the
proof is not the simplest, or the proof you would have given.
• C (partially correct), if the claim is correct and the proof
is largely a correct claim, but contains one or two incorrect
statements or justications.
• F (failure), if the claim is incorrect, the main idea of the
proof is incorrect, or...

A natural number p is a prime number provided that the only
integers dividing
p are 1 and p itself. In fact, for p to be a prime number, it is
the same as requiring that
“For all integers x and y, if p divides xy, then p divides x or p
divides y.”
Use this property to show that
“If p is a prime number, then √p is an irrational number.”
Please write down a formal proof.

Write a formal proof to prove the following conjecture to be
true or false.
If the statement is true, write a formal proof of it. If the
statement is false, provide a counterexample and a slightly
modified statement that is true and write a formal proof of your
new statement.
Conjecture: There does not exist a pair of integers m and n such
that m^2 - 4n = 2.

For each of the statements below, say what method of proof you
should use to prove them. Then say how the proof starts and how it
ends. Pretend bonus points for filling in the middle.
a. There are no integers x and y such that x is a prime greater
than 5 and x = 6y + 3.
b. For all integers n , if n is a multiple of 3, then n can be
written as the sum of...

True Or False
1. If nn is odd and the square root of nn is a natural number
then the square root of nn is odd.
2. The square of any even integer is even
3. The substraction of 2 rational numbers is rational.
4. If nn is an odd integer, then n2+nn2+n is even.
5. If a divides b and a divides c then a divides bc.
6. For all real numbers a and b, if a^3=b^3 then a=b.

Statement: "For all integers n, if n2 is odd then n is odd"
(1) prove the statement using Proof by Contradiction
(2) prove the statement using Proof by Contraposition

For each problem, say if the given statement is True or False.
Give a short justification if needed.
Let f : R + → R + be a function from the positive real numbers
to the positive real numbers, such that f(x) = x for all positive
irrational x, and f(x) = 2x for all positive rational x.
a) f is surjective (i.e. f is onto).
b) f is injective (i.e. f is one-to-one).
c) f is a bijection.

This problem outlines a proof that the number π is irrational.
Suppose not, Then there are relatively prime positive integers a
and b for which π = a/b. If p is any polynomial let Ip= ∫0a/bp(x)
sinx dx. i. Show that if p is non-negative and not identically 0 on
[0,a/b] then Ip>0; ii. Show that if p and all of its derivatives
are integer-valued at 0 and a/b then Ip is an integer. iii. Let N
be a large...

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 6 minutes ago

asked 16 minutes ago

asked 17 minutes ago

asked 25 minutes ago

asked 35 minutes ago

asked 45 minutes ago

asked 48 minutes ago

asked 51 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago