CSC165H1S Midterm 1
Duration: 75 minutes
代写数学表达与推理 This examination has 4 questions. All statements predicate logic must have negations applied directly topropositional variables or predicates.
- This examination has 4 questions.
- All statements predicate logic must have negations applied directly topropositional variables or predicates.
- You may not define your own propositional operators, predicates, or sets, unless asked to do so in the question.
Please work with the symbols we have introduced in lecture, and any additional definitions provided in the questions.
- Proofs should follow the guidelines used in the course (e.g., explicitly introduce all variables, clearly state all assumptions, justify every deduction in your proof body, etc.)
- In your proofs, you may always use definitions from the course. However, you may not use any external facts about these definitions unless the yare given in the question.
- You may not use induction for your proofs on this midterm.
1.[8 marks] Short answers questions. 代写数学表达与推理
(a) [2 marks] Let S1 be the set of all prime numbers, and S2 = {x | x ∈ N and x | 30}. Write down all the elements of S2\S1.
(b) [3 marks] Write down a truth table for the following expression in propositional logic. Rough work (e.g., intermediate columns of the truth table) is not required, but can be included if you want.
(¬p ⇔ q) ⇒ r
(c) [3 marks] Consider the following statement (assume predicates P and Q have already been defined):
∀x ∈ N, P(x) ⇒ (∃y ∈ N, Q(x, y))
Suppose we want to prove this statement. Write the complete proof header for a proof; you may write statements like “Let x = _____ ” without filling in the blank. The last statement of your proof header should be “We will prove that. . .” where you clearly state what’s left to prove, in the same style as the lectures or the Course Notes.
You do not need to include any other work (but clearly mark any rough work you happen to use).
2.[7 marks] Translations. 代写数学表达与推理
Let P be the set of all pets, and suppose we definethe following predicates:
- Cat(x): “x is a cat”, where x ∈ P
- Cute(x): “x is cute”, where x ∈ P
- Loves(x, y): “x loves y”, where x, y ∈ P (note that Loves(x, y) does not mean the same thing as Loves(y, x))
Translate each of the following statements into predicate logic. No explanation is necessary. Do not define any of your own predicates or sets. You may use the = and ≠ symbols to compare whether two pets are the same.
(a) [1 mark] Every cat loves itself.
(b) [2 marks] Every cat loves at least one pet that is cute.
(c) [2 marks] If at least one cat is cute, then every cat is cute.
(d) [2 marks] For every two distinct (i.e., not equal) pets, if the two pets love each other, then exactly one of them is a cat.
4.[5 marks] Divisibility. 代写数学表达与推理
Prove the following statement.
∀a, b ∈ N, b | a ∧ b | (a + 2) ⇒ b = 1 ∨ b = 2
Clearly state where you use any definition from class in your proof. We have left you space for rough work here and on the next page, but write your formal proof in the box below.
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