Question

) 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 | ≤ 2. (d) If x is a real number such that |x + 2| + |x| ≤ 5, then | x ^2 + 2x − 1 ≤ 2.|

Answer #1

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

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 .

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
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)

Prove or disprove the following statements.
a) ∀a, b ∈ N, if ∃x, y ∈ Z and ∃k ∈ N such that ax + by = k,
then gcd(a, b) = k
b) ∀a, b ∈ Z, if 3 | (a 2 + b 2 ), then 3 | a and 3 | b.

Prove or disprove:
If 2x ≡ 6 mod10, then x ≡ 3 mod10.

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...

5. Prove or disprove the following statements:
(a) Let R be a relation on the set Z of integers such that xRy
if and only if xy ≥ 1. Then, R is irreflexive.
(b) Let R be a relation on the set Z of integers such that xRy
if and only if x = y + 1 or x = y − 1. Then, R is irreflexive.
(c) Let R and S be reflexive relations on a set A. Then,...

8. Prove or disprove the following statements about
primes:
(a) (3 Pts.) The sum of two primes is a prime number.
(b) (3 Pts.) If p and q are prime numbers both greater than 2,
then pq + 17 is a composite number.
(c) (3 Pts.) For every n, the number n2 ? n + 17 is always
prime.

Prove or disprove: If a real 5x5 matrix has a non-real
eigenvalue, then it has 5 distinct eigenvalues.

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