Question

(b) If n is an arbitrary element of Z, prove directly that n is even iff n + 1 is odd. iff is read as “if and only if”

Answer #1

Perform the following tasks:
a. Prove directly that the product of an even and an odd number
is even.
b. Prove by contraposition for arbitrary x does not equal -2: if
x is irrational, then so is x/(x+2)
c. Disprove: If x is irrational and y is irrational, then x+y is
irrational.

3. Prove by contrapositive: Let n ∈ N. If n^3−5n−10>0,then n
≥ 3.
4. Prove: Letx∈Z. Then5x−11 is even if and only if x is odd.
4. Prove: Letx∈Z. Then 5x−11 is even if and only if x is
odd.

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)

Prove that any arbitrary function Ψ(x) can always be written as
a sum of an even and an odd function.

Prove the following: Let n∈Z. Then n2 is odd if and
only if n is odd.

Prove that n is prime iff every linear equation ax ≡ b mod n,
with a ≠ 0 mod n, has a unique solution x mod n.

Let x, y ∈Z. Prove that (x+1)y^2 is even if and only if x is odd
and y is even.

Prove: Let a and b be integers. Prove that integers a and b are
both even or odd if and only if 2/(a-b)

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

Let a,b ∈ Z. Prove that a−b is even if and only if x and y are
of the same parity.

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