Question

Definition of Even: An integer n ∈ Z is even if there exists an integer q ∈ Z such that n = 2q.

Definition of Odd: An integer n ∈ Z is odd if there exists an integer q ∈ Z such that n = 2q + 1.

Use these definitions to prove the following:

Prove that zero is not odd. (Proof by contradiction)

Answer #1

Discrete Math
6. Prove that for all positive integer n, there exists an even
positive integer k such that
n < k + 3 ≤ n + 2
. (You can use that facts without proof that even plus even is
even or/and even plus odd is odd.)

Let n be an integer. Prove that if n is a perfect square (see
below for the definition) then n + 2 is not a perfect square. (Use
contradiction) Definition : An integer n is a perfect square if
there is an integer b such that a = b 2 . Example of perfect
squares are : 1 = (1)2 , 4 = 22 , 9 = 32 , 16, · ·
Use Contradiction proof method

Using either proof by contraposition or proof
by contradiction, show that:
if n2 + n is irrational, then n is irrational.
Using the definitions of odd and even show that the
following 4 statements are equivalent:
n2 is odd
1 − n is even
n3 is odd
n + 1 is even

Prove the following theorem: For every integer n, there is an
even integer k such that
n ≤ k+1 < n + 2.
Your proof must be succinct and cannot contain more than 60
words, with equations or inequalities counting as one word. Type
your proof into the answer box. If you need to use the less than or
equal symbol, you can type it as <= or ≤, but the proof can be
completed without it.

Prove the following theorem: For every integer n, there is an even
integer k such that
n ≤ k+1 < n + 2.
Your proof must be succinct and cannot contain
more than 60 words, with equations or inequalities
counting as one word. Type your proof into the answer box. If you
need to use the less than or equal symbol, you can type it as <=
or ≤, but the proof can be completed without it.

Prove the following statements by contradiction
a) If x∈Z is divisible by both even and odd integer, then x is
even.
b) If A and B are disjoint sets, then A∪B = AΔB.
c) Let R be a relation on a set A. If R = R−1, then R is
symmetric.

Let n be any integer, prove the following statement:
n3+ 1 is even if and only if n is odd.

(a) Let N be an even integer, prove that GCD (N + 2, N) = 2.
(b) What’s the GCD (N + 2, N) if N is an odd integer?

Prove that for every positive integer n, there exists an
irreducible polynomial of degree n in Q[x].

Problem 2: (i) Let a be an integer. Prove that 2|a if and only
if 2|a3.
(ii) Prove that 3√2 (cube root) is irrational.
Problem 3: Let p and q be prime numbers.
(i) Prove by contradiction that if p+q is prime, then p = 2 or q
= 2
(ii) Prove using the method of subsection 2.2.3 in our book that
if
p+q is prime, then p = 2 or q = 2
Proposition 2.2.3. For all n ∈...

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