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Prove the following statement using induction. “For all positive integers n, 5|(n5 – n).”  Use good proof technique.  Note:  To avoid the need for typing superscript exponents, you may use the expression ‘n^5’ to represent n5. Grading rubric:1 pt. State the basis step, then prove it.1 pt. State the inductive hypothesis.2 pt. Complete the proof of the inductive step.1 pt. State the final conclusion at the end of the proof.1 pt. Label each part: the basis step, inductive hypothesis, inductive step, and conclusion.

Prove that 2×2 + x + 4 is O(x2), by identifying values for C and k and demonstrating that they do satisfy the definition of big-O for this function.  Show your work. Note:  To avoid the need for typing superscript exponents, you may use the notation ‘x^2′ to represent x2.

Define S, a set of integers, recursively as follows: Initial Condition: 0 ∈ SRecursion: If m ∈ S then m + 2 ∈ S. Which of the following sets is equivalent to set S?

The function f : ℝ ⟶ ℝ defined by f(x) = 5×3 + 3×2 – x + 7 is O(x).

Given relation R defined on the set { 2, 4, 6, 8 } as follows: (m, n) ∈ R if and only if m|n. Determine which properties relation R exhibits.  Select ‘True’ if the property does apply to relation R; otherwise select ‘False’.  There may be more than one or none. [A]   reflexive [B]   irreflexive [C]   symmetric [D]   antisymmetric [E]   asymmetric [F]   transitive

To complete the division algorithm equation, a = mq + r, using a = – 46 and m = 9, which of the following gives appropriate values for integers q and r, with r expressed as a non-negative integer between 0 and (m-1), inclusive.

For arbitrary positive integers a, b, and m with m>1, if a ≡ b (mod m), then a = b + km, for some integer k.

If the square root of every integer is an integer, then 2 is irrational.

For any predicates, P(x) and Q(x), ∀x [ P(x) ⋁ Q (x) ] ⟺ [ ( ∀x P(x) ) ⋁ ( ∀x Q(x) ) ].

For arbitrary positive integers a, b, c, and m with m>1, if (a + c) ≡ (b + d) (mod m), then a ≡ b (mod m) and c ≡ d (mod m).