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Math Help - Help with a logic question are they right?

  1. #1
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    Help with a logic question are they right?

    so i did these questions and was wondering if they are right since it is and even problem there is no solution in the back to know if they are right. my answers are at the bottom.

    let p,q, and r be the propositions
    p: you get an A on the final exam.
    q: you do every exercise in this book.
    r: you get an A in this class.

    write these propositions using p, q, and r and logical connectives.

    a) you get an A in this class, but you do not do every exercise in this book.

    b) you get an A on the final, you do every exercise in this book, and you get an A in this class

    c) to get an A in this class, it is necessary for you to get an A on the final.

    d) you get an A on the final, but you don't do every exercise in this book; nevertheless, you get an A in this class.

    e) getting an A on the final and doing every exercise in this book is sufficient for getting an A in this class.

    f) you will get an A in this class if and only if you either do every exercise in this book or you get an A on the final.

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  2. #2
    MHF Contributor
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    The answer to C is r -> p: if you get an A in the class, you know you must have got an A in the final because acing the final is necessary for A in the class; there is no way around it.
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  3. #3
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    Hello, camboguy!

    emakarov is absolutely correct!

    "Necessary" and "sufficient" can be confusing concepts.
    We must be very careful.

    I'll create an example.


    Suppose you always park your car in your uncovered driveway.

    p: It rains.
    q: Your car is wet.


    Then "If it rains, then your car is wet" is true: . p \to q
    . .
    Every time it rains your car gets wet.


    There are two interpretations of this implication.

    [1] p is a sufficient condition for q.
    . . .Knowing that it is raining, you know your car is wet.
    . . .
    You see that it's raining, you don't have to look at your car.

    [2] q is a necessary condition for p.
    . . .If it rains, your car must get wet.
    . . .
    A wet car is the result of rain.



    Back to your problem:

    p: You get an A on the final.
    r: You get an A in this class.

    (c) To get an A in this class it necessary for you to get an A on the final.


    \text{It says: }\:\underbrace{\text{A on final}}_p\text{ is a necessary condition for }\underbrace{\text{A in class}}_r

    This is the second interpretation . . . Therefore: . r \to p

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