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).

Which of these matrices is an adjacency matrix for K 1, 3 ?

What topic in units 10-11 of the course material did you find the most interesting?