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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).

Use the Euclidean algorithm to determine the GCD(268, 108).  Show your work. Then express the GCD(268, 108) value you identify as a linear combination of 268 and 108. Show your work.