Question

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

Answer #1

) Prove or disprove the statements: (a) If x is a real number
such that |x + 2| + |x| ≤ 1, then |x ^2 + 2x − 1| ≤ 2. (b) If x is
a real number such that |x + 2| + |x| ≤ 2, then |x^ 2 + 2x − 1| ≤
2. (c) If x is a real number such that |x + 2| + |x| ≤ 3, then |x
^2 + 2x − 1 |...

Here are two statements about positive real numbers. Prove or
disprove each of the statements
∀x, ∃y with the property that xy < y2
∃x such that ∀y, xy < y2 .

(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.

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...

Prove the following:
For any positive real numbers x and y, x+y ≥
√(xy)

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 <...

Prove the statement
For all real numbers x, if x − ⌊x⌋ < 1/2 then ⌊2x⌋ =
2⌊x⌋.

Discrete Math
5. Prove the following existential statements:
(a) There exists a real number x such that x2 −4x + 3
= 0.
(b) There is a real number x such that (x ≥ 1) → (x2
< 0)

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...

Prove the following using the specified technique:
(a) Prove by contrapositive that for any two real numbers,x and
y,if x is rational and y is irrational then x+y is also
irrational.
(b) Prove by contradiction that for any positive two real
numbers,x and y,if x·y≥100 then either x≥10 or y≥10.
Please write nicely or type.

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