Question

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 all positive integers n, there is an even integer k such that 2n < k < 2n + 3

(c) Use proof by contraposition to show that, “for all real numbers a, if a^2 + 3 is irrational, then a is irrational”.

Answer #1

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...

(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.

Consider the following statement:
If x and y are integers and x - y is odd, then x is odd or y is
odd.
Answer the following questions about this statement.
2(a) Provide the predicate for the starting assumption for a
proof by contraposition for the given statement.
2(b) Provide the conclusion predicate for a proof by
contraposition for the given statement.
2(c) Prove the statement is true by contraposition.
2(d) Prove that the converse is not true.

For Problems #5 – #9, you willl either be asked to prove a
statement or disprove a statement, or decide if a statement is true
or false, then prove or disprove the statement. Prove statements
using only the definitions. DO NOT use any set identities or any
prior results whatsoever. Disprove false statements by giving
counterexample and explaining precisely why your counterexample
disproves the claim.
*********************************************************************************************************
(5) (12pts) Consider the < relation defined on R as usual, where
x <...

Discreet Math: Prove or disprove each statement
a) For any real number x, the floor of 2x = 2 the floor of x
b) For any real number x, the floor of the ceiling of x = the
ceiling of x
c) For any real numbers x and y, the ceiling of x and the
ceiling of y = the ceiling of xy

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 the statement " For all real numbers r, if r is
irrational, then r/2 is irrational ". You may use any method you
wish. Be sure to state what method of proof you are using.

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.

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.

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) =...

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