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. [A]   K2 [B]   C3 [C]   Q4 [D]   W5

Consider the following problem:  1391(mod 11) = _________ Show how Fermat’s Little Theorem can be used to solve this problem. Express your answer as a non-negative integer less than the modulus. Note:  To avoid the need for typing superscript exponents, you may use the notation ‘x^n’ or the expression ‘x to the nth’ (with numbers in place of x and n), to represent xn.

Prove, or provide a counterexample to disprove, the following statement:             “The function f : ℝ ⟶ ℤ defined by f(x) = ⌊ 2x ⌋ 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 special symbols, instead of using the floor symbols in the function definition ⌊ 2x ⌋ you may use the expression ‘floor of ( 2x )’.

Use the Euclidean algorithm to determine the GCD(324, 147).  Show your work. Then express the GCD(324, 147) value you identify as a linear combination of 324 and 147.  Show your work.

Which of the following are built-in functions that perform I/O (input and output)?

There exists a simple graph with 2 vertices of degree 4 and 4 vertices of degree 3.

Which of these matrices is an adjacency matrix for K 2, 3 ?

Which of the following is not a valid expression? Assume x and y are integer variables.

For each positive integer k, let Ak = { x ∈ ℝ | 0  ≤  x  ≤ 1/k  }. Which of the following sets is equal to ?

The function f : ℝ ⟶ ℝ defined by f(x) = x3 + 3×2 – x + 7 is O(x4).