Question

If a , b ∈ N , then there exist unique integers q and r for which a = b q + r and 0 ≤ r < q .Is this statement true? If yes, give a proof of the statement. If not, give a counterexample.

Answer #1

**Proof:**

1. Let a ∈ Z and b ∈ N. Then there exist q ∈ Z and r ∈ Z with 0
≤ r < b so that a = bq + r.
2. Let a ∈ Z and b ∈ N. If there exist q, q′ ∈ Z and r, r′ ∈ Z
with 0 ≤ r, r′ < b so that a = bq + r = bq′ + r ′ , then q ′ = q
and r...

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...

. The distance between two nonempty sets A ⊆ R and B ⊆ R is
deﬁned as follows: d(A,B) = inf{|a − b| | a ∈ A,b ∈ B}.
(a) Find the distance between A ={1/ n | n ∈ N } and B ={1 − 1
/n | n ∈ N }
(b) Give a proof or a counterexample for the following
statement:
If d(A,B) > 0 then A ∩ B = ∅.
can someone please help me with...

1. Write a proof for all non-zero integers x and y, if there
exist integers n and m such that xn + ym = 1, then gcd(x, y) =
1.
2. Write a proof for all non-zero integers x and y, gcd(x, y) =
1 if and only if gcd(x, y2) = 1.

Write a formal proof to prove the following conjecture to be
true or false.
If the statement is true, write a formal proof of it. If the
statement is false, provide a counterexample and a slightly
modified statement that is true and write a formal proof of your
new statement.
Conjecture: There does not exist a pair of integers m and n such
that m^2 - 4n = 2.

Give an example of three positive integers m, n, and r, and
three integers a, b, and c such that the GCD of m, n, and r is 1,
but there is no simultaneous solution to
x ≡ a (mod m)
x ≡ b (mod n)
x ≡ c (mod r).
Remark: This is to highlight the necessity of “relatively prime”
in the hypothesis of the Chinese Remainder Theorem.

9. Let a, b, q be positive integers, and r be an integer with 0
≤ r < b. (a) Explain why gcd(a, b) = gcd(b, a). (b) Prove that
gcd(a, 0) = a. (c) Prove that if a = bq + r, then gcd(a, b) =
gcd(b, r).

(1) Determine whether the propositions p → (q ∨ ¬r) and (p ∧ ¬q)
→ ¬r are logically equivalent using either a truth table or laws of
logic.
(2) Let A, B and C be sets. If a is the proposition “x ∈ A”, b
is the proposition “x ∈ B” and
c is the proposition “x ∈ C”, write down a proposition involving a,
b and c that is logically equivalentto“x∈A∪(B−C)”.
(3) Consider the statement ∀x∃y¬P(x,y). Write down a...

Statement: "For all integers n, if n2 is odd then n is odd"
(1) prove the statement using Proof by Contradiction
(2) prove the statement using Proof by Contraposition

Let p and q be any two distinct prime numbers and define the
relation a R b on integers a,b by: a R b iff b-a is divisible by
both p and q. For this relation R: Show that the equivalence
classes of R correspond to the elements
of ℤpq. That is: [a] = [b] as equivalence
classes of R if and only if [a] = [b] as elements of
ℤpq.
you may use the following lemma: If p is prime...

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