COT 3100 Exam 1 PDF

Okay, here is the converted markdown format of the document. ### COT 3100 Spring 2024 Exam 1 **Name:** Brielle Ashmeade **U-Number:** 438424858 #### Instructions Read all instructions carefully. In order to receive full credit, correct answers must be well-organized and printed legibly and larg...

Okay, here is the converted markdown format of the document. ### COT 3100 Spring 2024 Exam 1 **Name:** Brielle Ashmeade **U-Number:** 438424858 #### Instructions Read all instructions carefully. In order to receive full credit, correct answers must be well-organized and printed legibly and large enough to read without straining. Give sufficient justification for questions as indicated. There are 120 points available on this exam, but it will be graded out of 100. As such, if you are struggling on a particular question or part, move on and come back if there is time. Now, keep your eyes on your own exam, turn off and do not use any electronics/communication device(s), and do not communicate with anyone other than a proctor during the examination. Remember to answer the last question on the back page. 1. **(6 points)** Identify the equivalence or rules of inference using the terms labeled a. through h. below. | | a. | b. | c. | d. | e. | f. | g. | h. | Equivalence or Rule of Inference | | :-- | :-: | :-: | :-: | :-: | :-: | :-: | :-: | :-: | :----------------------------- | | i) | | | | | | ⬤ | | | $p \land q \therefore q$ | | ii) | | | | | | | ⬤ | | $q \therefore p \lor q$ | | iii) | | | | | | ⬤ | | | $p \rightarrow q \\ q \rightarrow r \therefore p \rightarrow r$ | | iv) | ⬤ | ⬤ | | | | | | | $p \\ q \therefore p \land q$ | | v) | | | | | ⬤ | | | | $p \lor q \\ p \rightarrow r \\ q \rightarrow r \therefore r$ | | vi) | ⬤ | | | | | | | | $\sim(p \land q) \equiv \sim p \lor \sim q$ | a. Conjunction b. De Morgan's Law c. Generalization d. Elimination e. Division into Cases f. Specialization g. Transitivity h. Contradiction ### COT 3100 Spring 2024 Exam 1 2. **(6 points)** Which of the following should be considered statements? Make no assumptions about any unknowns. Select all that apply. * ✅ Today is Saturday * ✅ $42 = 49$ * ☐ They have a dog. * ☐ $12x = 49$ * ☐ Q(x, y) * ☐ This sentence is true. 3. **(8 points)** Complete the following truth table. | p | q | ~p | ~q | $p \rightarrow q$ | $q \rightarrow p$ | $q \land r$ | $p \lor \sim q$ | $q \rightarrow r$ | $p \land q$ | | :- | :- | :-: | :-: | :----------------: | :----------------: | :---------: | :-------------: | :-------------: | :---------: | | T | T | F | F | T | T | T | T | T | T | | T | F | F | T | T | F | T | T | F | F | | F | T | T | F | T | T | F | F | F | F | | F | F | T | T | T | T | T | T | T | F | 4. **(6 points)** Consider the statement: If Compound X is boiling, then its temperature is at least 150 degrees. Select all statements that are logically equivalent to the one above: ✅ If its temperature is less than 150 degrees, then Compound X is not boiling. ☐ If Compound X is not boiling, then its temperature is at least 150 degrees. ☑ Compound X is not boiling, or its temperature is at least 150 degrees. ☐ Compound X is boiling, or its temperature is less than 150 degrees. ☐ If its temperature is at least 150 degrees, then Compound X is boiling. ☐ If Compound X is not boiling, then its temperature is less than 150 degrees. ### COT 3100 Spring 2024 Exam 1 5. You are standing in the middle of a cornfield and hear: * If you build it, they will come. (a) (6 points) Determine how each statement below best relates to the above conditional. | | a. | b. | c. | d. | e. | f. | Statement | | :-- | :-: | :-: | :-: | :-: | :-: | :-: | :---------------------------------- | | | | | ⬤ | | | | If they will come, you'll build it. | | | | | | ⬤ | | | You'll build it, but they won't come. | | | | ⬤ | | | | | You won't build it, or they will come. | | | | | | ⬤ | | | If they won't come, you won't build it.| | | | | | | ⬤ | | If you won't build it, they won't come.| | | | | | | | ⬤ | If they will come, you won't build it.| a. Converse b. Inverse c. Contrapositive d. Equivalent disjunction e. Negation f. None of these (b) (6 points) Suppose it is true that they come. No other information is known. Determine the truth values of the statements below or indicate that the truth value cannot be determined from this information. | | True | False | Indeterminate | Statement | | :-- | :--: | :---: | :------------: | :----------------------------------------------- | | | | | ⬤ | If they will come, you'll build it. | | | | ⬤ | | You'll build it, but they won't come. | | | ⬤ | | | You won't build it, or they will come. | | | | ⬤ | | If they won't come, you won't build it. | | | | ⬤ | | If you won't build it, they won't come. | | | | | ⬤ | If they will come, you won't build it. | ### COT 3100 Spring 2024 Exam 1 6. **(6 points)** Negate each inequality as either another inequality, or conjunction/disjunction of inequalities. Your answers may only possibly include numbers, the words "and" or "or", or symbols ≤ or <. Note that this also excludes the use of & and 4. | Inequality | Negation | | :----------- | :--------- | | $-1 \le x$ | $x < -1$ | | $x < 2$ | $x \ge 2$ | | $-1 \le x < 2$ | $x < -1$ or $x \ge 2$ | 7. **(7 points)** Find an equivalent form of the expression $(\sim c \land a) \rightarrow \sim b$ that only makes uses of connectives ~ and $\land$ and the given variables. Show your work and state any equivalences used. $\sim[(\sim c \land a) \rightarrow \sim b]$ $(\sim c \land a) \land \sim(\sim b)$ Negation $(\sim c \land a) \land b$ Double negation $\sim c \land a \land b$ Associativity of $\land$ **Equivalent expression:** $\sim c \land a \land b$ ### COT 3100 Spring 2024 Exam 1 8. **(12 points)** Consider the following argument form: $q \land r \rightarrow p$ $r \lor \sim q$ $\sim q \rightarrow r$ $\therefore p$ (a) Complete the truth table for this argument form below and then select ☑ each box that corresponds to a critical row. Some space has been given to add as many helper columns as needed. | p | q | r | $\sim q$ | $q \land r$ | $q \land r \rightarrow p$ | $r \lor \sim q$ | $\sim q \rightarrow r$ | p | | :- | :- | :- | :-------: | :----------: | :-----------------------: | :------------: | :-------------------: | :- | | T | T | T | F | T | T | T | T | T | | T | T | F | F | F | T | F | T | T | | ☑ | T | F | T | F | T | T | T | T | | F | T | T | F | T | F | T | T | F | | ☑ | F | T | T | F | T | T | T | T | | F | F | T | T | F | T | T | T | F | | ☑ | F | F | T | F | T | T | T | T | | F | F | F | T | F | T | T | T | F | (b) Is the given argument form valid? Select ◯ Yes, or ⬤ No. Your answer must follow correctly from your work above to receive credit. ### COT 3100 Spring 2024 Exam 1 9. **(8 points)** A syllogism has the following argument form: Major Premise: $p \rightarrow q$ Minor Premise: $\sim q$ Conclusion: $\therefore \sim q$ Consider the major premise: If life exists on other planets, then we are not alone in the universe. Now consider each minor premise/conclusion pair below that could form an argument with this major premise. If the resulting argument is invalid, then simply write "invalid" in the box. If the resulting argument is valid, state a correct name for the the argument form. | | Minor Premise | Conclusion | Form | | :-- | :------------------------------- | :------------------------------- | :--------------- | | (a) | We are not alone in the universe | Life exists on other planets | Invalid | | (b) | We are alone in the universe | Life does not exist on other planets | Modus Tollens | | (c) | Life does not exist on other planets | We are alone in the universe | Invalid | | (d) | Life exists on other planets | We are not alone in the universe | Modus Ponens | 10. **(6 points)** Consider statement forms $(p \lor q) \rightarrow r$ and $(p \rightarrow r) \land (q \rightarrow r)$. Are these statement forms logically equivalent? ◯ Yes, or ☐ No. To receive credit, justify your answer below. Two (2) extra points will be given for a correct justification that does not rely on a truth table. $(p \rightarrow r) \land (q \rightarrow r)$ $=(\sim p \lor r) \land (\sim q \lor r)$ De Morgan's Law $=(\sim p \land \sim q) \lor r$ Distributive $=\sim (p \lor q) \lor r$ De Morgan's law $=(p \lor q) \rightarrow r$ De Morgan's Law ### COT 3100 Spring 2024 Exam 1 11. Suppose that the following premises are true. Then, for each part below, deduce the truth of the given statement as a sequence of arguments using valid rules of inference. You may not use a truth table, nor make any assumptions that you have not deduced from these premises. State each rule of inference used and reference premises and previous deductions appropriately. Note that t is a variable, not to be confused with a tautology t. $\sim q \land t \rightarrow \sim s$ (P1) $q \rightarrow r$ (P2) $s \lor \sim p$ (P3) $\sim r$ (P4) $\sim q \lor p \rightarrow t$ (P5) (a) (3 points) Deduce the truth of $\sim q$ from the given premises. This should require one step. $q \rightarrow r$ (P2) $\sim r$ (P4) $\therefore \sim q$ Modus Tollens (b) (6 points) Deduce the truth of t from the given premises and any of your (correct) results from part (a). This should require two steps. Step 1: $\sim q$ (part a) $\therefore \sim q \lor p$ (Generalization) Step 2: $\sim q \lor p \rightarrow t$ (P5) $\therefore \sim q \vor p$ (above) $\therefore t$ (Modus ponens) (c) (6 points) Deduce the truth of $\sim s$ from the given premises and any of your (correct) results from parts (a) and (b). This should require two steps. Step 1: $\sim q$ (part a) $t$ (part b) $\therefore \sim q \land t$ (Conjunction) Step 2: $\sim q \land t \rightarrow \sim s$ (P1) $\therefore \sim q \land t$ (Step1) $\therefore \sim s$ (Modus ponens) (d) (3 points) Deduce the truth of $\sim p$ from the given premises and any of your (correct) results from parts (a), (b), and (c). This should require one step. $s \lor \sim p$ (P3) $\therefore \sim s$ (part c) $\therefore \sim p$ (Elimination) ### COT 3100 Spring 2024 Exam 1 12. (6 points) For each part below, consider the quantified statement $\forall x \in X, \exists y \in Y: \sim Q(x,y)$ (*) Note: For part (b), do not assume the condition given in (a), i.e. consider (a) and (b) separately. (a) If $X\neq 0$, what can be said about the truth value of (*)? *It is true. *It is false. *Its truth value cannot be determined without more information. (b) Suppose that (*) is false. Which of the following are then true? Select ☑ all that apply. *▢ Q(x, y) *▢ $\forall x \in X, \sim[\exists y \in Y : Q(x, y)]$ *☑ $\sim[\forall x \in X, \exists y \in Y: \sim Q(x, y)]$ *☐ $\exists x \in X: \forall y \in Y, \sim Q(x, y)$ *☑$\exists x \in X: \forall y \in Y, Q(x,y)$ *☐ No $x \in X$ satisfies that there is some $y \in Y$ such that $Q(x, y)$. 13. (6 points) A strange island contains two types of natives, knights who always tell the truth and knaves who always lie. As you arrive, you are approached by three natives A, B, and C. A says: If B is a knave, then C is a knave. B says: A is a knight. C says: A is a knave. Which scenarios below are possible? Select☑ all that apply. * ☐ A, B, and C are knights. * ☐✅ A and B are knights, C is a knave. * ☑ A is a knight, B and C are knaves. * ☐ A and C are knights, B is a knave. * ☐ B is a knight, A and C are knaves. * ☐B and C are knights, A is a knave. * ☐ C is a knight, A and B are knaves. * ☐ A, B, and C are knaves. ### COT 3100 Spring 2024 Exam 1 14. (8 points) Real numbers are Archimedean in the sense that For every $ \forall y \in R, \exists x \in Z_+ $ such that if $q \land r y > 0$ then $p x < y$. As such, the following negation is false: There exists $y \in R$ such that for all $x \in Z_+, y>0 but \frac{1}{x} \ge y$ Show that this negation is false by writing (a) in symbolic form, applying the simple negation ~ to that symbolic form, and use your understanding of negations of quantified statements to express ~(a) in a symbolic form equivalent to (b). Show the steps to obtain this form by working one quantifier at a time, as we did in class. $\forall y \in R, \exists x \in Z_+ : if y > 0 then \frac{1}{x} < y$ $ \sim[\forall y \in R, \exists x \in Z_+ : if y > 0 then \frac{1}{x} < y]$ $\exists y \in R, \sim[\exists x \in Z_+ : if y > 0 then \frac{1}{x} < y]$ $\exists y \in R, \forall x \in Z_+, \sim [ if y > 0 then \frac{1}{x} < y]$ $\exists y \in R, \forall x \in Z_+, y>0 by \frac{1}{x} \ge y$ ### COT 3100 Spring 2024 Exam 1 You may use the extra space here for scratch work, but do not tear off any pages. 15. (5 points) Write down your seat row (A-Z) and number (1-100). Seat letters may be found at the end of each row and numbers on your seat. Confirm each. Don't guess. Seat numbers may be found on your seat. Row letters may be found at the end of each row. Seat row letter: G Seat number: 4 Now, acknowledge that you have adhered to the academic integrity policy for this course and USF by signing below. Without your correct name, signature, seat row, and seat number, your exam will be graded as zero until formally meeting with the instructor. Regardless, no points will be awarded for this question if not answered correctly during the exam. **Brielle Ashmeade** Print your Name **B Ashmeade** Signature