Could some other math teacher take a look at this?

Discussion in 'General Education' started by Aliceacc, Jun 9, 2012.

  1. Aliceacc

    Aliceacc Multitudinous

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    Jun 9, 2012

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  3. Aliceacc

    Aliceacc Multitudinous

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    It must be a very busy day for math teachers.

    In that case, I'll stand by my more-or-less on target explanation-- hope it helps.
     
  4. swansong1

    swansong1 Virtuoso

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    I looked at that post. Makes me REALLY glad I'm not a math teacher! So...I'll just say, have wonderful afternoon!
     
  5. MikeTeachesMath

    MikeTeachesMath Devotee

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    It's been a really busy day for me, sorry! I've been up and going since 7. Finally got a chance to sit down.

    I flipped through the entire chapter and nowhere did my professor outline the laws of equivalence like you did. Yours make so much more sense.

    So, let me see if I understand this.

    Law of Detachment:
    p......q.......p->q
    T......T.......T
    F......T.......F
    F......F.......T
    T......F.......F

    Disjunctive Inference:
    p......q.......p v q
    T......T.......T
    T......F.......T
    F......F.......F
    F......T.......T

    Are those truth tables correct?

    Also, the proof you posted makes more sense than anything I've done so far, so I'm pretty sure you're right. And I can follow it pretty well. The interesting thing is you started by manipulating the left side of the equivalence. My professor, when I asked for help, said to start by manipulating the right side...

    I guess I just need to adapt. I keep forgetting that this class is all about "switching gears" from asking "What is 2 + 2?" to "WHY is 2 + 2? What is 2? What is addition?"

    It's absolutely beautiful but absolutely frustrating for the time being. I'm sure I'll get the hang of it.

    Thanks so much Alice and MathEqualsLove :)
     
  6. Aliceacc

    Aliceacc Multitudinous

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    Get the Sequential III book I recommended on the other thread, Mike. It does a great job with the laws of logic as I recall.
     
  7. MathEqualsLove

    MathEqualsLove Companion

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    Okay, you've got a slight problem in your truth table for the Law of Detachment. I remember being so confused by this when I took this class. On the second line if p is false and q is true, then p implies q is true. This means a statement such as: If 2+2=6, then 3+2=5 is true.

    I googled around for a more eloquent explanation than what I could came up with, and I found this: "If p is false, then the implication with p as the hypothesis will not meet its condition (that p be true) so q does not have to be either true or false. Either way, the implication has not been denied, because its condition was not met, so the implication stands as true. "

    Your truth table for the disjunctive inference is perfect.
     
  8. Aliceacc

    Aliceacc Multitudinous

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    Here's the rule for Conditional statements:

    If the premise is true and the conclusion is true, the statement is true.
    "If you study, then you'll pass." You study. You pass. The statement was true.

    If the premise is true and the conclusion is false, the statement is false.
    "If you study, then you'll pass." You study. You don't pass. The statement was false.

    If the premise is false, it doesn't matter about the conclusion-- the statement is assumed to be true. (Kind of like innocent until proven guilty.)
    "If you study, then you'll pass." You don't study. Maybe you pass, maybe you don't. Maybe you cheat. Maybe everyone passes. The original statment cannot logically be proven false, so it's true.
     
  9. Aliceacc

    Aliceacc Multitudinous

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    And Disjunctions:

    For starters, a Disjunction is a statement using the word "Or."
    For example, "I'll have an appetizer or a dessert."
    If the first part is true (I have an appetizer) the statement is true.

    If the second part is true (I have a dessert) the statement is true.

    If both parts are true (I have both an appetizer and a dessert) the statement is true.

    The only way a disjuction is false is if BOTH parts are false-- I opt not to have an appetizer and I opt not to have a dessert.

    Conjunctions:
    A conjunction is an "AND" statement. For example:
    I teach Geometry and Algebra.
    If both parts are true, the statement is true.

    I teach Geometry. I teach Algebra. The statement is true.

    If either part is false, the statement is false.
    I don't teach Geometry. The statement is false.
    I don't teach Algebra. The statement is false.

    If both parts are false, the statement is false.
    I teach neither Geometry nor Algebra. The statement is false.

    In other words, the only way a conjunction can be true is if BOTH pieces are true.
     
  10. chebrutta

    chebrutta Enthusiast

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    Ooooh, logic. That was the only part of math that really made sense to me... probably because I could substitute statements and figure it out.

    Your explanation made sense to me, Alice.
     
  11. Aliceacc

    Aliceacc Multitudinous

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    Oops, forgot biconditionals--- see what happens when you don't see something for 20 years????

    A biconditional is a statement of the "If and only if" form; the symbol is a double arrow.

    "I get a raise if and only if I get a promotion." It's another way of saying that the raise brings the promotion and the promotion brings the raise.

    Basically, it's true only when both parts have the same truth value. It's a conjunction, so both parts have to match.



    A tautology is a statement which is always true, so the final column of the truth table must be all true.

    I think that's all the rules for the truth tables-- does it help at all?

    Now get a list of all the rules; I'm not sure my list was complete. Have them in front of you as you do the logic proofs.

    Once you start playing with them, you'll see that they're not that awful.

    For some examples, look at old Course II Regents in the NY State library. There was always a choice of proofs-- 2/3 where the choices were logic, coordinate and geometric. So most of the old Regents should have a logic proof if you just hunt around a bit. You'll find a lot of truth tables in the old Course I Regents; there was just about ALWAYS a truth table on the Part II.
     
  12. MikeTeachesMath

    MikeTeachesMath Devotee

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    Yes! It helped so much, thank you. I submitted the assignment last night so we'll see what happens when he gets back to me today. Crossing my fingers.
     
  13. MikeTeachesMath

    MikeTeachesMath Devotee

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    I'm seething. I got a 70 (14/20), but I lost points in ridiculous ways.

    I lost 2 points on a sets question because I didn't express a set in the "best" way. I got no credit for the question even though my answer was still correct.

    For the proof, he said that I didn't prove anything and instead just ended up with the same formula I started with and that my answer is hardly useful. Alice, this is the question you did. He also said that one of the properties Alice used wasn't listed in our text therefore can't be used, because apparently I'm only supposed to be referring to the test and no outside sources.

    But what REALLY gets me is his answer for the proof:

    (p implies r) and (q implies r)
    = (not(p) or r) and (not(q) or r)
    = (apply a distributive property)
    = ...
    = (p or q) implies r

    Didn't he do the same thing we did, except start from the other side?!?!

    I lost 2 points because I stated that two sets cannot be represented by a disjoint union of sets. In his text, he states that if two sets have elements in common, they cannot be expressed as a disjoint union. Now he's telling me they can.

    What the ****.
     
  14. Aliceacc

    Aliceacc Multitudinous

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    There's that one reason I was unsure of. It exists, but I'm unsure of the name of it.

    But the other reasons I used, along with the proof I did, would have been acceptable on a NY State Regents exam. Unless it somehow got lost in the translation, I started off with the statement on the left side and ended up with the statement on the right side... two different statements.

    I would check his textbook for clues as to which side must be simplified, and why.

    The set notation is pretty particular; I'm willing to bet that it's a fine line, but an important one that cost you those 2 points.

    On the disjoint thing, did you mention the facts that the sets can't have any elements in common?
     
  15. MikeTeachesMath

    MikeTeachesMath Devotee

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    I checked the chapter over and over again. Nothing in the way of "do this side first".

    Plus, he did the exact same thing as you did. He started with one side and manipulated the other until they were identical. So I don't understand why it's wrong.

    This class is going to put such a damper on my summer.
     
  16. Mathemagician

    Mathemagician Groupie

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    If you didn't mention that the sets have a common element, then you made a fatal mistake. You have to be so precise in these courses that it hurts.

    On the other hand, I've taken college math courses where 70%=A due to awesome curves so not all is lost.
     
  17. MikeTeachesMath

    MikeTeachesMath Devotee

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    I did say that!

    "2Z U (1 + 3Z) = {..., -6, -5, -4, -2, 0, 1, 2, 4, 6, ...} This set cannot be expressed as the union of disjoint sets because 2Z and (1 + 3Z) have elements in common."

    So I don't know. Hopefully he'll curve.
     
  18. MikeTeachesMath

    MikeTeachesMath Devotee

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    Okay, I feel like an idiot. I can't get this.

    Let p represent "The Buffalo Bills scored more than 20 points"

    Let q represent "The Buffalo Bills win the game"

    Find the truth-value for p -> q and not p -> not q in the following table:

    p.........q.........p->q.........not p -> not q
    T.........T
    T.........F
    F.........T
    F.........F

    Okay, so the first one. P -> q where P is true and Q is true. So, "If the Buffalo Bills score more than 20 points then the Buffalo Bills win the game". I guess? I know nothing about sports... so true?

    So now not p -> not q where P is true and Q is true. So, "If the buffalo bills don't score more than 20 points then the Buffalo Bills don't win the game." Again, no sports knowledge. I'm assuming they don't win. So true again?

    Then p -> q where P is true and Q is false. "If the buffalo bills score more than 20 points then they don't win the game". I'm assuming this is false. So not p -> not q: "If the buffalo bills don't score more than 20 points then they do win the game". Assuming this is also false.

    I feel like I'm not doing this right at all.

    I finished the truth table:

    p.........q.........p->q.........not p -> not q
    T.........T..........T.................T
    T.........F...........F................F
    F.........T..........F.................T
    F.........F..........T.................T

    Alice, I ordered that textbook you recommended.
     
  19. Aliceacc

    Aliceacc Multitudinous

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    It has nothing to do with sports, it's about the truth value of the premise and of the conclusion.

    Go back to the rules I gave you for a conditional statement. It's true all the time EXCEPT when the first part is true and the second part is false.

    Those rules need to be either understood or memorized. Print up those posts, highlight the rules, and put them in front of you.

    And maybe 5:40 am isn't the best time to do this?

    The Course II text should be a huge help on the proofs. Course I covers truth tables, but honestly the notes I gave you should probably hit the high points.
     
  20. MikeTeachesMath

    MikeTeachesMath Devotee

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    OH, the same rules apply to every scenario? That makes so much more sense.

    And yeah, normally math at 6 AM doesn't happen, but I have a really busy day and this is the only time I can fit some homework in.
     
  21. Aliceacc

    Aliceacc Multitudinous

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    Yeah, the same rules apply. So if you want to mentally change p to " I studied hard for the test" and q to "I got a great grade" it won't effect the truth table at all.
     
  22. MikeTeachesMath

    MikeTeachesMath Devotee

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    Oh. Duh.

    I swear my professor's text would better serve me as a doorstop or paperweight. I can't wait for that other text to get here. I emailed my AP Calc teacher and he said they got rid of the Sequence texts ages ago, otherwise he would have given me one.
     
  23. Aliceacc

    Aliceacc Multitudinous

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    Ok, now go back and redo the truth table, including the columns you're missing. I won't give you the answers on this, but it's a whole lot easier when you include all columns you need.

    Truth tables need to be built, one tiny piece at a time. Each component of the final statement needs its own column. So, for example, if there's a "~p" anywhere in the statement, you'll need a "~p" column. If there's a "pV ~q", then you'll need columns for p, q (obviously, right?? They're in every single one) ~q, then pV~q. Each time, you're looking at one (for negations) or two other columns to determine the truth values of each line.

    Hope that helps. I'm off in a few minutes to rally the troops for school.
     
  24. Aliceacc

    Aliceacc Multitudinous

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    Yeah, the couse was a nightmare; we dropped it a good 20 years ago. (Hence my memory loss on the name of that one rule.) I seriously doubt that we have any copies at school. But the text, particularly the chapter on logic, was great. In fact, just about ANY Keenan/Dressler text is good; if you see some along the way at garage sales, pick them up!
     
  25. MikeTeachesMath

    MikeTeachesMath Devotee

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    Perfect score on my second assignment. Thanks so much Alice :).

    I love Keenan/Dressler texts. After my junior year is when Math A and B were done away with, so of course I asked one of my former teachers for the Keenan/Dressler Math A and Math B textbooks. I still use them for reference when I need refreshing on something basic.
     
  26. Aliceacc

    Aliceacc Multitudinous

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    You're welcome.

    Keenan and Dressler are both local guys-- one of them taught in Bayside and the other in NYC. One, I forget which, died a few years ago, but the other is still writing. I love their work-- they teach math the same way I do, with lots of solid explanations and lots of practice problems.
     
  27. MikeTeachesMath

    MikeTeachesMath Devotee

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    Didn't one of them teach at Pace for a while as well?

    I would like to know which one of them picked that blinding red color for the Math B textbook though :lol:.
     
  28. Aliceacc

    Aliceacc Multitudinous

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    Yeah, Dressler taught at Pace. (I have one of their Algebra/Trig books out as luck would have it.) I would love to have had either of them as a classroom teacher.

    I'm guessing the cover design was Amsco's choice. Course III was Neon Green, before neon became popular. You could spot that book a mile away, particularly if a kid was doing his math homework in someone else's class :)
     

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