Complete your choice of one of the proofs given below.  PROOF 1: Prove the following statement using a proof by cases.   [Hint: there are 3 cases] “For all positive integers n with 2 ≤ n ≤ 4, n!/2 ≤ 2n.” Use good proof technique.   Grading rubric:1 pt. State any givens and assumptions.3 pt. Clearly identify the cases and prove each case.1 pt. State the final conclusion at the end of the proof. Note:  Remember that n factorial, written as n!, is defined as n(n-1)…(2)1, the product of n times every positive integer less than n.   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

For each positive integer k, let Ak = { x ∈ ℝ | 1/k  ≤  x  ≤  k  }. Which of the following sets is equal to ?

Use the Euclidean algorithm to determine the GCD(357, 156).  Show your work. Then express the GCD(357, 156) value you identify as a linear combination of 357 and 156.  Show your work.

All of the following are methods scientists use to classify bacteria EXCEPT:

Determine which of these set identities are supported by the entries in the membership table given below.  There may be more than one or none. Select ‘True’ if the identity is supported by this given membership table; otherwise select ‘False’. [1]  (A – B) – C ⊆ (A – B) [2]  (A – C) ≠ (A – C) – B [3]  (A – B) ⊂ (A – C) – B [4]  (A – C) – B ⊄ (A – B) [5]  (A – C) ⊈ (A – B) [6]  (A – B) – C = (A – C) – B A B C A – C A – B (A – C) – B (A – B) – C 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 1 0 0 1 1 0 1 0 0 0 1 1 1 0 0 0 0

The function f : ℕ ⟶ ℕ defined by f(n) = 3n2(log n) is O(n2log n).

Which one of the following structures would NOT be found in a prokaryotic cell?

Prove, or provide a counterexample to disprove, the following statement:             “The function f : ℤ ⟶ ℕ defined by f(n) = n2 is a bijection.” Use good proof technique.  Remember that a bijection is both one-to-one (injective) and onto (surjective).  To prove, you must demonstrate both properties are true; to disprove, you only need a counterexample that shows one of the properties is not valid. Grading rubric:1 pt.  Indicate whether you will be proving or disproving the assertion.  Also, if proving, state both definitions, one-to-one and onto; if disproving, state the definition you plan to 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.

What is the function of the fungal fruiting body?

The main reason reptiles are so well-adapted to a land environment is their method of