Question

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 consecutive integers.

c. For all integers a and b , if a^2 + b^2 is odd, then a or b is odd.

Discrete Stuctures 1

Answer #1

Use only ∀,∃,¬,∧,∨,=,!= to translate the following statement
into a ﬁrst-order logical formula. (You are NOT allowed to use any
other symbols like →,>,<, etc.)
S(n) = “The number n cannot be written as the sum of three or
more consecutive positive integers.”
Let n be an odd number greater than 1. Prove that n is a prime if
and only if S(n) in (i) is true.

Use the method of direct proof to prove the following
statements.
26. Every odd integer is a difference of two squares. (Example 7
= 4 2 −3 2 , etc.)
20. If a is an integer and a^ 2 | a, then a ∈ { −1,0,1 }
5. Suppose x, y ∈ Z. If x is even, then x y is even.

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 the following sentences as quantified logical statements, using the universal and existential quantifiers, and defining predicates as needed.
Second, write the negations of each of these statements in the same way.
Finally, choose one of these statements to prove. If it is true, prove it, and if it is false, prove its negation. Your proof need not use symbols, but can be a simple explanation in plain English.
1. If m and n are positive integers and mn is...

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...

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...

Please note n's are superscripted.
(a) Use mathematical induction to prove that 2n+1 +
3n+1 ≤ 2 · 4n for all integers n ≥ 3.
(b) Let f(n) = 2n+1 + 3n+1 and g(n) =
4n. Using the inequality from part (a) prove that f(n) =
O(g(n)). You need to give a rigorous proof derived directly from
the definition of O-notation, without using any theorems from
class. (First, give a complete statement of the definition. Next,
show how f(n) =...

When we say Prove or disprove the
following statements, “Prove” means you show the
statement is true proving the correct statement using at most 3
lines or referring to a textbook theorem.
“Disprove” means you show a statement is wrong by
giving a counterexample why that is not true).
Are the following statements true or not? Prove or disprove
these one by one. Show how the random variable X looks in each
case.
(a) E[X] < 0 for some random...

Multiple Choice
Select the best answer from the available choices for each
question.
Which of the following is NOT part of the definition of
a sample space S?
S can be discrete or continuous
Each outcome must be in S at most once
Each element in S is equally likely
Each outcome must be in S at least once
S is a set of possible outcomes in an experiment
Three A’s, three B’s, and two C’s are arranged at
random...

What tools could AA leaders have used to increase their
awareness of internal and external issues?
???ALASKA AIRLINES: NAVIGATING CHANGE
In the autumn of 2007, Alaska Airlines executives adjourned at
the end of a long and stressful day in the
midst of a multi-day strategic planning session. Most headed
outside to relax, unwind and enjoy a bonfire
on the shore of Semiahmoo Spit, outside the meeting venue in
Blaine, a seaport town in northwest
Washington state.
Meanwhile, several members of...

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