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 ⟺ .

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.

For arbitrary positive integers a, b, c with a ≠ 0, if a | (b + c) then a | b or a | c.

Indicate which of these listed graphs are bipartite. Select ‘True’ if the graph is bipartite; otherwise select ‘False’.  There may be more than one or none.    K4    C6    Q3    W5

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

Prove the following statement using induction. “For all integers n ≥ 3, 2n + 1 ≤ 2n.” Use good proof technique.  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. Note: To avoid the need for typing superscript exponents, you may use the expression ‘2^n’ to represent 2n.  Also the ≥ symbol can be written as >=.

The function f : ℝ ⟶ ℝ defined by f(x) = 5×3 + 3×2 – x + 7 is O(x4).

Indicate a reason for each assertion in the argument below. Choose your answers from the given list of Rules of inference and logical equivalences.  An item from the list may be used as a reason more than once. Assertion Reason Premise 1:   r  ˄ ( ¬s → ¬t  ) Given Premise 2:    t Given A.      r B.      ( ¬s → ¬t  ) C.      ( t  → s ) D.      s E.       s  ˄  r