Prove, or provide a counterexample to disprove, the following statement:             “The function f : ℕ ⟶ ℕ  be defined by f(n) = n2 + 3 is onto.” Use good proof technique. Grading rubric:1 pt. State the definition of onto at the beginning, then prove or disprove.1 pt. State any givens and assumptions.1 pt. Clearly explain your reasoning.1 pt. Remember to state the final conclusion at the end of the proof. Note:  To avoid the need for typing superscript exponents, you may use the expression ‘n^2’ or ‘n-squared’ to represent n2.

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.    reflexive    irreflexive    symmetric    antisymmetric    asymmetric    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.

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

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

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