Homework 1
Due: Friday, 9/13/2024 before 11:59PM ET
Total Points: 50 (45+ 5 pts extra credit)
Write Up Instructions
Submit your completed homework to Brightspace electronically in PDF format. Any submissions that are not a PDF or not a legible PDF will not receive credit. We need to be able to read your submission to be able to grade your work.
Your write-up should contain enough information from the problem so that a reader doesn’t need to return to the text to know what the problem is (it is a good habit to rewrite each problem prior to solving it). There is no general rule for how much information from the problem to include, but it should be possible to read your homework and ascertain what the problem was and what your solution is accomplishing. When writing up the solution, you may hand write the solutions and submit a scanned PDF copy. If you hand write your solutions, make sure that you write clearly, and your writing is legible. Double check your scans to make sure that your scanned copy is legible.
After you submit your work, make sure the file is visible. Download your submitted copy, open it, and see whether you submitted the correct file and whether your submitted file has not been corrupted during the upload. You can upload your submission multiple times. Only the last file will be graded. Keep in mind that if your completed work consists of multiple pages and you submit a separate file for each page, only the last file submitted will be graded. In this case, only one page of your submission would be graded.
Be careful about potential plagiarism. Your submitted work must be your own.
Late Policy
Late homework will be accepted up to 1 day late for a penalty of 25% of the total points. For example, if the homework is worth 100 points and you submit it one day late, you will receive the maximum of (your score earned minus 25 points) and 0 points.
Assigned Problems
Problem 1. (5 pts = 1×5) State whether each of the following sentences is a proposition or not. Then, if the sentence is a proposition, write its negation. Otherwise, provide a brief justification why the sentence is not a proposition.
a) Submit your homework on time.
b) Is it raining today?
c) 5 + x = 10.
d) Sky is pink.
e) 10 + 11 = 25.
Problem 2. (5 pts = 1×5) Assume the propositions p, q, r, and s have the following truth values:
• p : False (F). • q : True (T). • r : False (F). • s : True (T)
Evaluate the truth value for each of the following compound propositions. Show your work for your evaluation.
a) p ⊕ q Λ r
b) (p ⊕ q) Λ r
c) pv q ↔ q → p
d) ¬pv q ⊕ r → s Λ ¬q
e) ¬p v (¬q ⊕ r) → (s Λ ¬q)
Problem 3. (5 pts = 4 + 1) Answer the following questions.
a) Construct a truth table for the expression (p ⊕ q) → (¬p ⊕ ¬q).
b) For the following truth table, give a logical expression whose truth table is the same as the one given.
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p
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q
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?
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F
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F
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T
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F
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T
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T
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T
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F
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T
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T
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T
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F
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Problem 4. (10 pts = 2×5) Consider the following propositions:
• p : You share your solutions with your classmates.
• q : You get an academic integrity violation case against you.
• r : You pass the course.
For each of the following problem, provide the corresponding English sentence.
a) p → q
b) q → ¬r
c) ¬(q Λ r).
d) (p Λ q) v r
e) ¬r ↔ (p ^ q)
Problem 5. (10 pts = 2×5) Consider the following true propositions:
• h : John is healthy.
• w : John is wealthy.
• s : John is wise.
For each of the following sentences write, symbolically, the compound proposition that corresponds to the given sentence in English as it is written (do not change the order or form of the expression).
a) John is healthy and wealthy but not wise.
b) John is not wealthy but he is healthy and wise.
c) John is neither healthy, wealthy, nor wise.
d) John is neither wealthy nor wise, but he is healthy.
e) John is wealthy, but he is not both healthy and wise.
Problem 6. (5 pts) Using a truth table determine whether (¬p v q) ↔ (p ^ ¬q) is a tautology, or a contradiction, or a contingency.
Problem 7. Using Equivalence Laws, show that -
a) (5 pts) ¬(p v (¬p ^ q)) is logically equivalent to ¬p ^ ¬q.
b) (Extra Credit)(5 pts) ¬(p v ¬( p ^ q )) is logically equivalent to F (i.e. a contradiction).