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

For all sets A, B, and C, if A ⊆ B and A ⊆ C then A ⊆ B ⋃ C.

Given relation R defined on the set { 5, 10, 15, 20 } 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

According to Fermat’s little theorem, what is the value of 7352(mod 11)?

The decimal expansion of the hexadecimal number 7B16 is _______________ten. Only type the digits; do not include the base.

Prove the following statement using a proof by contradiction.  “For all real numbers x and y, if 8x + 6y = 211, then either x is not an integer or y is not an integer.” Use good proof technique.  Grading rubric: 1 pt. State what is given and what is assumed true to begin the proof. 1 pt. Explain your steps, including identifying the contradiction that is reached. 1 pt. State the final conclusion of the proof.

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:  r ˄ t Given A.      r B.      ¬ r ∨ ( ¬s ∨ ¬t ) C.      ( ¬s ∨ ¬t  ) D.      t E.       ¬s

Which of these is a diagram for K 2, 2 ?

Prove the following statement using induction. “For all integers n ≥ 4, 2n < n !"  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: 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 >=.

Prove the following statement by proving the contrapositive.  “If n2 + 3  is odd then n is even, for all n ∈ ℤ.” Use good proof technique.  Grading rubric:1 pt. State the contrapositive at the beginning, then prove it.1 pt. State any givens and assumptions.1 pt. Clearly explain your reasoning.1 pt. 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-squared’ or ‘n^2’ to represent n2.