Consider proving the following statement using a proof by cases. “For all integers n, n ≤ n2.” What 3 cases do you use for this proof?  What do you demonstrate must be true to complete the proof of each case? 

For all sets A and B, if sets A and B are not disjoint, A – B = A.

A ____ connects all the computers in a system.

Compared to a low ratio grid, a high ratio grid will1.  absorb more primary radiation2.  absorb more scattered radiation3.  allow less centering latitude

Prove the following statement using induction. “For all integers n ≥ 2, n2 ≥ n+1. ”  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: To avoid the need for typing superscript exponents, you may use the expression ‘n^2’ to represent n2.  Also the ≥ symbol can be written as >=.

Prove the following statement using a direct proof. “For all integers m and n, if m is odd and n is even, then (m n) is even.” Use good proof technique.   Grading rubric:1 pt. State what is given and assumed true to begin.1 pt. Clearly explain your steps.1 pt. State the final conclusion at the end of the proof.

For all sets A, B, and C, A ⋂ (B ⋃ C) = (A ⋃ B) ⋂ (A ⋃ C).

Consider proving the following statement by proving the contrapositive. “If (n – 1) is odd, n is even, for all n ∈ ℤ.” What do you assume as true to begin the proof?  What do you demonstrate must be true to complete the proof? 

The membership table in the previous problem shows that the two given sets equal. 

Prove the following statement using a proof by cases.   “For all non-negative integers n ≤ 2, 2n ≤ 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:  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