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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?   [Assume]  What do you demonstrate must be true to complete the proof?   [Prove]  

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.

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 [1] [2] [3] [4] 0 0 1 [5] [6] [7] [8] 0 1 0 [9] [10] [11] [12] 0 1 1 [13] [14] [15] [16] 1 0 0 [17] [18] [19] [20] 1 0 1 [21] [22] [23] [24] 1 1 0 [25] [26] [27] [28] 1 1 1 [29] [30] [31] [32]  

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 [1] [2] [3] [4] [5] 0 0 1 [6] [7] [8] [9] [10] 0 1 0 [11] [12] [13] [14] [15] 0 1 1 [16] [17] [18] [19] [20] 1 0 0 [21] [22] [23] [24] [25] 1 0 1 [26] [27] [28] [29] [30] 1 1 0 [31] [32] [33] [34] [35] 1 1 1 [36] [37] [38] [39] [40]  

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), 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) (C – B) A – (B – C) A – (C – B) 0 0 0 [1] [2] [3] [4] 0 0 1 [5] [6] [7] [8] 0 1 0 [9] [10] [11] [12] 0 1 1 [13] [14] [15] [16] 1 0 0 [17] [18] [19] [20] 1 0 1 [21] [22] [23] [24] 1 1 0 [25] [26] [27] [28] 1 1 1 [29] [30] [31] [32]  

Prove the following statement using a proof by contradiction. “If the sum of 6 different integers is greater than 42, then at least one of the numbers must be greater than 7.” Use good proof technique.  Grading rubric:1 pt. State what is assumed true to begin the proof. 1 pt. Clearly explain your steps. Identify the contradiction that is reached. 1 pt. State the final conclusion of the proof.

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 )’.