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

Complete this proof by identifying the statement that correctly matches each step in the inductive proof of this assertion, “For all integers n ≥ 4, n ! ≥ n2.”

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

For all sets A and B, A ⊆ (A ⋂ B).

Prove the following statement using a proof by cases.   [Hint: there are 3 cases] “For all positive integers n ≤ 3, n! ≤ n2+1 .” 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:  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 ‘n-squared’ or ‘n^2’ to represent n2.  Also the ≤ symbol can be written as

The function f : ℤ+ ⟶ ℝ + defined by f(x) = 1/x2 is surjective (onto).

Prove, or provide a counterexample to disprove, the following statement:             “The function f : ℝ ⟶ ℤ, defined by f(x) = ⌊ x + 2 ⌋  is onto.” Notice the use of the floor function in the definition of function f.  Use good proof technique. Grading rubric:1 pt. State the definition of onto 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 special symbols, instead of using the floor symbols in the function definition ⌊ x + 2 ⌋ you may use the expression ‘floor of ( x + 2 )’.

If 2 is the square of a positive integer, 0 > 1.

Let Ak  = { x ∈ ℝ | k-1 ≤  x  ≤ k }, for each positive integer k.  What is ?

For all sets A and B, if A ⊂ B, then A ∩ B = A.