Question

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.

Answer #1

Prove or disprove following by giving examples:
(a) If X ⊂ Y and X ⊂ Z, then X ⊂ Y ∩ Z
(b) If X ⊆ Y and Y ⊆ Z, then X ⊆ Z
(c) If X ∈ Y and Y ∈ Z, then X ∈ Z

(§2.1) Let a,b,p,n ∈Z with n > 1.
(a) Prove or disprove: If ab ≡ 0 (mod n), then a ≡ 0 (mod n) or
b ≡ 0 (mod n).
(b) Prove or disprove: Suppose p is a positive prime. If ab ≡ 0
(mod p), then a ≡ 0 (mod p) or b ≡ 0 (mod p).

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,...

(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.

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...

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 (a) Z[x]/(x^2 + 1), (b) Z[x]/(x^2 - 1) is an
Integral domain.
By showing (a) x^2+1 is a prime ideal or showing x^2 + 1 is not
prime ideal.
By showing (b) x^2-1 is a prime ideal or showing x^2 - 1 is not
prime ideal.
(Hint: R/I is an integral domain if and only if I is a prime
ideal.)

) 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 |...

Prove or disprove the following statement: 2^(n+k) is an element
of O(2^n) for all constant integer values of k>0.

Prove the following statements:
1- If m and n are relatively prime,
then for any x belongs, Z there are integers a; b such that
x = am + bn
2- For every n belongs N, the number (n^3 + 2) is not divisible
by 4.

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