Question

Exercise 2.5.1: Proofs by cases.

Prove each statement. Give some explanation of your answer

(b)

If x and y are real numbers, then max(x, y) + min(x, y) = x + y.

(c)

If integers x and y have the same parity, then x + y is even.

The parity of a number tells whether the number is odd or even. If x and y have the same parity, they are either both even or both odd.

(d)

For any real number x, |x| ≥ 0.

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...

PLEASE DON'T USE ANY OTHER PROOFS FROM POOFS BY CASES! USE
DEFINITION OF ABSOLUTE VALUE!
And I was thinking of doing 4 cases:
1. x greater than or equal to 0, and y greater than
or equal to 0
2. x greater than or equal to 0, y less than 0
3. x less than 0, y greater than or equal to 0
4. x less than 0 , and y is less than 0.
Proofs by cases - absolute value.
Prove...

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...

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...

either prove that it’s true by explicitly using limit laws, or
give examples of functions that contradict the statement
a) If limx→0 [f(x)g(x)] exists as a real number, then both
limx→0 f(x) and limx→0 g(x) must exist as real numbers
b) If both limx→0 [f(x) − g(x)] and limx→0 f(x) exist as real
numbers, then limx→0 g(x) must exist as a real number

DISCRETE MATHS [ BOOLEANS AND LOGIC] Please answer
all
Exercise 1.7.1: Determining whether a quantified statement about
the integers is true.
infoAbout
Predicates P and Q are defined below. The domain of discourse is
the set of all positive integers.
P(x): x is prime
Q(x): x is a perfect square (i.e., x = y2, for some
integer y)
Indicate whether each logical expression is a proposition. If
the expression is a proposition, then give its truth value.
(c)
∀x Q(x)...

In class we proved that if (x, y, z) is a primitive Pythagorean
triple, then (switching x and y if necessary) it must be that (x,
y, z) = (m2 − n 2 , 2mn, m2 + n 2 ) for some positive integers m
and n satisfying m > n, gcd(m, n) = 1, and either m or n is
even. In this question you will prove that the converse is true: if
m and n are integers satisfying...

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...

2. Which of the following is a negation for ¡°All dogs are
loyal¡±? More than one answer may be correct.
a. All dogs are disloyal. b. No dogs are loyal.
c. Some dogs are disloyal. d. Some dogs are loyal.
e. There is a disloyal animal that is not a dog.
f. There is a dog that is disloyal.
g. No animals that are not dogs are loyal.
h. Some animals that are not dogs are loyal.
3. Write a...

PLEASE READ THE ARTICLES ATTACHED AND ANSWER THE
FOLLOWING QUESTION. THE ARTICLES ARE BOTH LISTED
PLEASE PROVIDE DETAILED EXPLANATIONS.
PLEASE WRITE ONE REFLECTION COMBINING BOTH
ARTICLES.
The purpose of the Article Reflection is to deepen your
engagement with the topic of Epidemiology. It will give you the
opportunity to reflect on the current real-life epidemiological
issues at hand and help to bring meaning to them.
ARTICLE 1:
A group of students knew they had covid-19. They hosted
a party over...

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