Question

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.

Answer #1

Determine if each of the following statements is true or false.
If a statement is true, then write a formal proof of that
statement, and if it is false, then provide a counterexample that
shows its false.
1) For each integer a there exists an integer
n such that a divides (8n +7) and
a divides (4n+1), then a divides 5.
2)For each integer n if n is odd, then 8
divides (n4+4n2+11).

1. For each statement that is true, give a proof and for each
false statement, give a counterexample
(a) For all natural numbers n, n2
+n + 17 is prime.
(b) p Þ q and ~ p Þ ~ q are NOT logically
equivalent.
(c) For every real number x
³ 1, x2£
x3.
(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...

Write a formal proof where you define a function to prove that
the sets 3Z and 6Z have the same cardinality, meaning |3 Z| = |6
Z|. 3Z = {..., −3, 0, 3, 6, 9, ...} and 6Z = {..., −6, 0, 6, 12,
18, ...}
Z = integers

For each of the following statements: if the statement is true,
then give a proof; if the
statement is false, then write out the negation and prove that.
For all sets A;B and C, if B n A = C n A, then B = C.

4. Let f be a function with domain R. Is each of the following
claims true or false? If it is false, show it with a
counterexample. If it is true, prove it directly from the FORMAL
DEFINITION of a limit.
(a) IF limx→∞ f(x) = ∞ THEN limx→∞ sin (f(x)) does
not exist.
(b) IF f(−1) = 0 and f(1) = 2 THEN limx→∞ f(sin(x)) does not
exist.

1. Write a proof for all non-zero integers x and y, if there
exist integers n and m such that xn + ym = 1, then gcd(x, y) =
1.
2. Write a proof for all non-zero integers x and y, gcd(x, y) =
1 if and only if gcd(x, y2) = 1.

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...

Discrete math problem!
Prove or disprove the following statement:
“If two rectangles have the same area and the same perimeter,
then they have the same dimensions (length, width).”
Note that finding a pair of rectangles that meet the criteria
and showing that their dimensions are the same is an example, not a
proof. If you feel that the statement is false, then demonstrate it
by showing that it leads to a contradiction, or by finding a
counterexample.

(a) Is the converse of Bolzano-Weierstrass Theorem true? If yes
prove it. If false provide a counterexample.
(b) Since Q is countably infinite, it can be written as a sequence
{xn}. Can {xn} be monotone? Briefly
explain. Hint. Assume it’s monotone, what would be the
consequences?
(c) Use the , N definition to prove that if {xn} and {yn} are
Cauchy then {xn + yn} is Cauchy too.

5. Determine whether the following statements are TRUE or FALSE.
If the statement is TRUE, then explain your reasoning. If the
statement is FALSE, then provide a counter-example. a) The
amplitude of f(x)=−2cos(X- π/2) is -2 b) The period of
g(x)=3tan(π/4 – 3x/4) is 4π/3.
. c) If limx→a f (x) does not
exist, and limx→a g(x) does not exist, then limx→a (f (x) + g(x))
does not exist. Hint: Perhaps consider the case where f and g are
piece-wise...

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