Question

(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.

Answer #1

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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...

3.a) Let n be an integer. Prove that if n is odd, then
(n^2) is also odd.
3.b) Let x and y be integers. Prove that if x is even and y is
divisible by 3, then the product xy is divisible by 6.
3.c) Let a and b be real numbers. Prove that if 0 < b < a,
then (a^2) − ab > 0.

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 that there are infinitely many primes of the form 3n+2,
where n is a nonnegative integer.

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

Let n be a positive odd integer, prove gcd(3n, 3n+16) = 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.

Prove or disprove that 3|(n^3 − n) for every positive integer
n.

Use Mathematical Induction to prove that 3n < n! if n is an
integer greater than 6.

Part #1:
Prove or disprove (formally or informally): The sum of an
integer and its cube is even.
Part #2:
Provide counterexamples to the following statements.
If n2 > 0 then n > 0.
If n is an even number, then n2 + 1 is prime.
(n2 is n to the power of 2).

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