Consider proving the following statement using a direct proof. “If a and b are real numbers with b > a, then b > (a+b)/2.” What do you assume as true to begin the proof?  What do you demonstrate must be true to complete the proof? 

The function f : ℤ+ ⟶ ℕ defined by f(x) = ⌊ x – 1 ⌋ is a bijection.

Prove the following statement using a proof by cases.   “For all non-negative integers n ≤ 2, n2 ≤ 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 ‘n-squared’ or ‘n^2’ to represent n2.  Also the ≤ symbol can be written as

Consider proving the following statement using a proof by contradiction. “The sum of an irrational number and rational number is irrational.” What do you assume as true to begin the proof?     What do you demonstrate must be true to complete the proof?    

For all sets A and B, ∅ ⊆ (A ⋂ B).

Let Ak  = { x ∈ ℝ | k-1 ≤  x  ≤ k }, for each positive integer k.   What is , where n is an arbitrary integer ≥ 2?

If A = { a, b, c } and B = { b, { c }}, then | ???? (A ⋃ B) | = 16.

Prove, or provide a counterexample to disprove, the following statement:             “The function f : ℕ ⟶ ℕ  be defined by f(n) = n2 + 5  is one-to-one.” Use good proof technique. Grading rubric: 1 pt. State the definition of one-to-one at the beginning, then prove or 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 superscript exponents, you may use the expression ‘n^2’ or ‘n-squared’ to represent n2.

Given arbitrary sets A, B, and C, complete the given membership table to verify whether the two sets, (A – B) ⋂ C and (A – C) ⋂ (B – C), are equal. Tip:  For any cells of the table that display beyond the right border of the question box, use the TAB →| key to move from cell to cell, rather than ‘clicking’ in a cell to make an entry. A B C (A – B) (A – C) (B – C) (A – B) ⋂ C (A – C) ⋂ (B – C)   0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1  

Given arbitrary sets A, B, and C, complete the given membership table to verify whether the two sets, A – (B – C) and C – (B – A), are equal. Tip:  For any cells of the table that display beyond the right border of the question box, use the TAB →| key to move from cell to cell, rather than ‘clicking’ in a cell to make an entry. A B C (B – C) (B – A) A – (B – C) C – (B – A) 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1