Logical form and Equivalence

📝 Practice Quiz: Logical Form & Equivalence

Part 1: Logical Form

  1. Rewrite the following argument into symbols, then identify its logical form:

    • If Maria is an artist or Maria is a musician, then Maria is creative. Maria is an artist. Therefore, Maria is creative.

  2. Construct an argument with completely different content but the same logical form as in (1).

Part 2: Statements

  1. Which of the following are statements? Explain why.
    a. "7 is greater than 2."
    b. "Close the door."
    c. "x² = 9."
    d. "What time is it?"

Part 3: Compound Statements

  1. Let p = "It is raining," q = "It is windy," r = "The power goes out."
    Write the following sentences symbolically:
    a. It is not raining but it is windy.
    b. It is neither raining nor windy.
    c. If it is raining or windy, then the power goes out.

  2. Rewrite the ambiguous expression:

    • p ∧ q ∨ r
      Write it two different ways with parentheses to make the meaning clear.

Part 4: Translation Practice

  1. Let h = "It is hot," s = "It is sunny." Translate:
    a. It is not hot and not sunny.
    b. It is sunny but not hot.
    c. If it is hot or sunny, then I will go swimming (let w = "I will go swimming").

Part 5: Inequalities as Logic

Let p = "0 < x," q = "x < 5," and r = "x = 5."
7. Write each of the following symbolically:
a. 0 < x < 5
b. x ≤ 5
c. 0 < x ≤ 5

🔍 Bonus Challenge

  1. Fill in the blanks so the following argument has the form: If p or q, then r. q. ∴ r.

    • If ____ or ____, then I will pass the class.

    • Therefore, I will pass the class.

👉 Do you want me to also create an answer key with step-by-step solutions (like your textbook examples), or would you rather try the quiz first and then check afterward?

You said:

i didnt get to 8 but ill put my answers here and grade me and give feedback on what i need help with and what i did good. 1. p V q -> c. p = if maria is a artist, V = or, q = If maria is a musician, ->c = then maria is creative, maria is an artist so p = true therefore c = true. 1. If snow is white and its cold, then its december. The snow isnt white. therefor its not december. 3. a = true 7 is always greateer than 2 in this example, b = true, c = false because it depends on what x's value is, d = true. 4. a ~p ^ q, b .~p V ~q, c. (pVq)->r. 5. (p and q) (q or r). 6. a. ~h^~s, b. s^~h, c. hVs->w. 7. p<q, q<r, didnt get the last one

ChatGPT said:

Great work! Let’s go through each of your answers step by step. I’ll grade, give corrections where needed, and explain what you did well.

1. Logical Form (Maria Argument)

Your answer:

  • You wrote it symbolically as: p ∨ q → c (with p = Maria is an artist, q = Maria is a musician, c = Maria is creative).

  • You also explained: since p = true, therefore c = true.

✅ Feedback: Excellent! You correctly identified the form If p or q, then r. p. ∴ r. (you used "c" instead of "r," but that’s fine). You also understood the reasoning flow.

Your second argument (different content):

  • If snow is white and it’s cold, then it’s December. The snow isn’t white. Therefore, it’s not December.

❌ Issue: This does not match the same form. The original was:

If p ∨ q, then r. q. ∴ r.
But yours is:
If p ∧ q, then r. ¬p. ∴ ¬r.

That’s a different structure (closer to modus tollens but incorrectly applied).

✅ What you did good: You created a valid-looking argument.
❌ What to fix: Make sure you keep the connective (or) the same. For example:

  • If it rains or I bring an umbrella, then I won’t get wet. I bring an umbrella. Therefore, I won’t get wet.

3. Statements

Your answers:
a. True → ✅ correct.
b. True → ❌ correction: "Close the door" is a command, not a statement (statements must be true or false).
c. False → ❌ correction: “x² = 9” is a statement form, but it depends on x’s value, so we say "not a statement until x is specified."
d. True → ❌ correction: “What time is it?” is a question, not a statement.

✅ What you did good: You recognized that some depend on variables.
❌ What to fix: Remember: statements must be declarative sentences that can be true or false. Commands and questions don’t count.

4. Compound Statements

Your answers:
a. ¬p ∧ q → ✅ correct! ("not raining but windy")
b. ¬p ∨ ¬q → ❌ correction: "Neither raining nor windy" = ¬p ∧ ¬q, not ∨.
c. (p ∨ q) → r → ✅ correct!

✅ What you did good: a and c were perfect.
❌ What to fix: Be careful: "neither…nor…" always means and with both negated.

5. Ambiguity in p ∧ q ∨ r

Your answer:

  • You wrote: (p and q) (q or r).

❌ This looks incomplete. The correct two possible interpretations are:

  • (p ∧ q) ∨ r

  • p ∧ (q ∨ r)

✅ What you did good: You recognized parentheses are needed.
❌ What to fix: Be precise in writing both full options.

6. Translation with h = hot, s = sunny

Your answers:
a. ¬h ∧ ¬s → ✅ correct.
b. s ∧ ¬h → ✅ correct.
c. (h ∨ s) → w → ✅ correct.

🎉 Perfect!

7. Inequalities (p = 0 < x, q = x < 5, r = x = 5)

Your answers:

  • a. p < q (you wrote this, but it should be p ∧ q).

  • b. q < r (you wrote this, but it should be q ∨ r).

  • c. Didn’t finish (should be p ∧ (q ∨ r)).

✅ What you did good: You noticed these need symbolic form.
❌ What to fix: Instead of writing "p < q," use logical connectives (∧, ∨). The solutions are:
a. p ∧ q
b. q ∨ r
c. p ∧ (q ∨ r)

📊 Your Grade & Feedback

  • 1a: ✅ Excellent

  • 1b: ❌ Wrong form

  • 3: 1/4 correct

  • 4: 2/3 correct

  • 5: ❌ Needs completion

  • 6: ✅ Perfect

  • 7: ❌ Needs practice

Overall: 65% (C range).

✅ Strengths:

  • You’re good at translating English into symbolic logic (Questions 1a, 4a, 4c, 6 all correct).

  • You understand how "or" and "and" work in many cases.

⚠ Needs Improvement:

  • Recognizing what counts as a "statement" (commands/questions don’t).

  • Keeping the logical form exactly the same when making new examples.

  • Being precise with "neither…nor…" (always ∧ with both negated).

  • Using ∧/∨ correctly in inequalities (instead of < or plain wording).

👉 Do you want me to make you a shorter practice set just on your weak spots (statements, neither/nor, and inequalities)?

You said:

yes please

ChatGPT said:

Perfect 👍 I’ll build you a short focused practice quiz targeting the areas you need the most practice on:

  • Statements (commands/questions vs. true/false)

  • Neither/Nor translations

  • Inequalities with ∧ and ∨

🎯 Focused Practice Quiz

Part A: Statements

For each sentence, say whether it is a statement (True/False) or not a statement (command/question/depends on variable).

  1. The moon has craters.

  2. Please close the book.

  3. x – 5 = 10

  4. What is your favorite color?

  5. 2 + 2 = 5

Part B: Neither/Nor

Let p = “I am tired” and q = “I am hungry.”
Translate into symbols:

  1. I am neither tired nor hungry.

  2. It is not the case that I am tired, and it is not the case that I am hungry.

  3. I am not tired but I am not hungry.

Part C: Inequalities

Let p = “0 < x,” q = “x < 4,” and r = “x = 4.”
Translate into symbols:

  1. 0 < x < 4

  2. x ≤ 4

  3. 0 < x ≤ 4

👉 Do you want me to also include an answer key right under this quiz (so you can check yourself instantly), or do you want to try it blind first and then I grade like last time?

You said:

A. 1. True it is a statement. 2.False its a question. 3. Im confused with this because in the lecture notes it says a statement is only a statement if it is either true or false not both. so from that i have to say false because x - 5 doesnt equal 10 if x doesnt equal 15. 4. false this is a question. 5. true. hmm technically its false but it is a statement. part B. 6. ~p ^ ~q. 7.~p^~q. 8. ~pV~q. part c 9.p^q. 10. qVr, 11. p^(qVr)

ChatGPT said:

Awesome — thanks for working through these! Let’s grade and walk through each carefully.

✅ Grading & Feedback

Part A: Statements

  1. The moon has craters. → Correct! This is a statement (true).

  2. Please close the book. → ❌ Careful: This is a command, not a question. So it’s not a statement.

  3. x – 5 = 10 → ❌ Correction: This is not a statement because its truth depends on the value of x. You were close, but the reasoning needs to be: not a statement, since it can’t be judged true/false until x is specified.

  4. What is your favorite color? → ✅ Correct: Not a statement (question).

  5. 2 + 2 = 5 → ✅ Correct: This is a statement (false, but still a statement).

Score: 3/5
👉 You’re improving! Just remember: commands/questions ≠ statements, and variable-dependent expressions are not statements until x has a value.

Part B: Neither/Nor

  1. I am neither tired nor hungry. → Your answer: ¬p ∧ ¬q → ✅ Correct!

  2. It is not the case that I am tired, and it is not the case that I am hungry. → ¬p ∧ ¬q → ✅ Correct!

  3. I am not tired but I am not hungry. → Your answer: ¬p ∨ ¬q → ❌ Correction: "but" works like "and." It should be ¬p ∧ ¬q.

Score: 2/3
👉 You nailed "neither/nor." Just watch out: "but" almost always means ∧.

Part C: Inequalities

  1. 0 < x < 4 → Your answer: p ∧ q → ✅ Correct!

  2. x ≤ 4 → Your answer: q ∨ r → ✅ Correct!

  3. 0 < x ≤ 4 → Your answer: p ∧ (q ∨ r) → ✅ Correct!

Score: 3/3 🎉 Perfect on this section!

📊 Overall Results

  • Part A (Statements): 3/5

  • Part B (Neither/Nor): 2/3

  • Part C (Inequalities): 3/3
    Final Score: 8/11 → ~73% (Solid C+/B–)

🌟 What You Did Well

  • You’re very strong on inequalities now — perfect translations.

  • You handled "neither/nor" correctly in most cases.

  • You’re good at identifying false but still valid statements (like 2 + 2 = 5).

⚠ What To Improve

  1. Statements (biggest gap):

    • A statement = must be declarative and capable of being true/false.

    • Commands and questions are not statements.

    • Equations with variables (like x – 5 = 10) are not statements until x is given.

  2. "But" vs. "Or":

    • "But" = logical and (∧), just adds contrast in English.

    • Don’t confuse it with "or."

👉 Want me to now give you a mini-drill just on Statements (commands/questions/variable cases) since that’s your weakest section?

You said:

no but you could tell me how to use these questions and answers so i can upload the into knowt so i can have them on flashcards with the correct answers

ChatGPT said:

Got it ✅ — you want to take these Q&A sets and turn them into flashcards in Knowt so you can study. Here’s the easiest way to do it:

🔹 Step 1: Format Questions & Answers

Knowt flashcards work best if each card has a front (question) and back (answer).
So, take our practice questions and put them into this format:

Example (from Part A):

  • Front (Q): Is "The moon has craters" a statement?

  • Back (A): Yes, it’s a statement (true).

🔹 Step 2: Use a Simple Q/A Layout

Here’s how your set would look if we rewrite it for flashcards:

Part A: Statements

  1. Q: Is "The moon has craters" a statement?
    A: Yes, it’s a statement (true).

  2. Q: Is "Please close the book" a statement?
    A: No, it’s a command, not a statement.

  3. Q: Is "x – 5 = 10" a statement?
    A: No, it depends on the value of x (not a statement until x is given).

  4. Q: Is "What is your favorite color?" a statement?
    A: No, it’s a question, not a statement.

  5. Q: Is "2 + 2 = 5" a statement?
    A: Yes, it’s a statement (false, but still a statement).

Part B: Neither/Nor

  1. Q: Translate "I am neither tired nor hungry" (p = tired, q = hungry).
    A: ¬p ∧ ¬q

  2. Q: Translate "It is not the case that I am tired, and it is not the case that I am hungry."
    A: ¬p ∧ ¬q

  3. Q: Translate "I am not tired but I am not hungry."
    A: ¬p ∧ ¬q

Part C: Inequalities

  1. Q: Translate "0 < x < 4" (p = 0 < x, q = x < 4, r = x = 4).
    A: p ∧ q

  2. Q: Translate "x ≤ 4."
    A: q ∨ r

  3. Q: Translate "0 < x ≤ 4."
    A: p ∧ (q ∨ r)

🔹 Step 3: Upload to Knowt

  1. Go to Knowt.com → Flashcards.

  2. Click “Create Flashcard Set.”

  3. Copy/paste each Q/A pair into the flashcard maker.

    • Put the Q on the front.

    • Put the A on the back.

  4. Save your set — Knowt will also let you turn them into a quiz/test later.