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

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 [Step3] B.      ¬ r ∨ ( ¬s ∨ ¬t ) [Step4] C.      ( ¬s ∨ ¬t  ) [Step5] D.      t [Step6] E.       ¬s [Step7]

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.

Which of these is a diagram for K 1, 4 ?

Consider the following proposition. “If no questions are dumb, then this question is smart.” Which of the following is the inverse of this statement? Only one statement is correct.  Note:  smart = not dumb

Define S, a set of integers, recursively as follows: Initial Condition: 0 ∈ SRecursion: If m ∈ S then 2m + 1 ∈ S. Which of the following sets is equivalent to set S?

With the universe of discourse for x as the set of all people alive in the world and the universe of discourse for y as the set of all countries in the world, we define the following predicates: F(x) is “x is a current FSU student,” G(x) is “x is a graduate of FSU,” and R(x, y) is “x is a resident of y.” Which of the following logical expressions accurately expresses this statement: Some residents of the United States are graduates of FSU and some residents of the United States are not graduates of FSU.