CSC 165 H1 — Question 1 of 4
数学表达与推理代写 1.[6 marks]For both statements below:(i)Write the negation of the original statement without using the ¬ symbol.
Submission Instructions
- You may type your answers or hand-write them legibly, on paper or using a tablet and stylus.
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For all questions in this test, “proof” means a formal proof that includes a header, and a proof body with justififications for each deduction. Each question can be answered correctly in less than one (1) page, and some questions have very short correct answers. You will NOT be penalized directly if you use more space for your answer, but longer answers increase the chance of errors… Remember that we are looking for evidence that you understand the conventions for writing correct proofs—NOT just how to answer these specifific questions—so pay close attention to the structure of your answers!
1.[6 marks] 数学表达与推理代写
For both statements below:
(i)Write the negation of the original statement without using the ¬ symbol.
(ii)Write whether the original statement is true or false.
(iii)If the original statement is true, prove it. If the original statement is false, disprove it.
(Note: The notation R≥0 represents the set [0, ∞) = {x ∈ R | x ≥ 0}.)
(a)∀x ∈ R, ∀n ∈ Z,(x < n) ⇒ (3x > 7n − 4)
(b)∀x ∈ R≥0, ∃n ∈ Z,(x ≤ n) ∧ (6 + 10x > 3n2 + 3)
Don’t forget: this test contains four separate questions (plus the Academic Integrity statement)!
2.[5 marks] 数学表达与推理代写
This question tests you on “proof by induction”. Even if there is a simple proof using another technique, you will receive at most half the marks if you do NOT use a proof by induction.
In your answer, you may use the facts that e = 2.71828 . . . and 1/e = 0.367879 . . . Also, if you want to look at the graph of any function in this question, you may use https://www.desmos.com/calculator—but NO other online resource is allowed.
Use induction to prove the following statement. As part of your answer, make sure to provide an explicit defifinition for your predicate P(n), and to state clearly what you are proving in each section of your proof.
∀n ∈ N,(n ≥ 2) ⇒ (e1-n + 3 < n2 + 2)
3.[3 marks] 数学表达与推理代写
This question tests you on “proof by contradiction”. Even if there is a simple proof using another technique, you will receive at most half the marks if you do NOT use a proof by contradiction.
Defifinition: A Pythagorian prime p is a prime number for which ∃d ∈ N, p = 4d + 1.
For example, 5 is a Pythagorean prime because 5 = 4(1) + 1.
Give a proof by contradiction that 11 is NOT a Pythagorean prime.
4. [5 marks]
In this question, you must use the following defifinition of absolute value:
Prove that every solution to
|x − 6| ≤ b − 2x
belongs to the set (−∞, b − 6], where b is the last non-zero digit in your student number. (Here, “last” means furthest to the right; for example, if your student number is 1000305070, the last non-zero digit is “7”.)
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