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Math Help - Validity Using Euler Circles and Truth Tables

  1. #1
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    Validity Using Euler Circles and Truth Tables

    I'm so confused on how to tackle this problem:



    1. Truth tables are related to Euler circles. Arguments in the form of Euler circles can be translated into statements using the basic connectives and the negation as follows:



    Let p be “The object belongs to set A. “Let q be “the object belongs to set B.”


    All A is B is equivalent to p -> q.

    No A is B is equivalent to p ->~ q.

    Some A is B is equivalent to p ^ q.

    Some A is not B is equivalent to p ^ ~q.


    Determine the validity of the next arguments by using Euler circles, then translate the statements into logical statements using the basic connectives, and using truth tables, determine the validity of the arguments. Compare your answers.


    (a). No A is B.
    Some C is A.
    ___________
    Therefore Some C is not B.


    (b) All B is A.
    All C is A.
    __________
    Therefore All C is B.
    Last edited by kma27; July 30th 2008 at 12:02 PM. Reason: Wanted to make my post easier to read.
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  2. #2
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    I have never seen “Euler circles” used in this context.
    It is well known that the nineteenth century English mathematician Venn used circles to illustrate the validity of certain syllogisms.

    Your first problem corresponds to the first diagram below.
    The A & B circles do not intersect: No A is B.
    The x in both A & C indicates existence: Some C is A.
    Having diagramed the two hypotheses, we can see that the conclusion is also diagramed: Some C is not B.

    On the other hand, the second diagram shows that the argument in the second problem is not valid. The two hypotheses could conceivably be diagramed in a way that the conclusion need not follow.

    BTW: Where did you get the term Euler circles?
    Attached Thumbnails Attached Thumbnails Validity Using Euler Circles and Truth Tables-venn1.gif   Validity Using Euler Circles and Truth Tables-venn2.gif  
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  3. #3
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    It's the term that our professor uses; though I've heard that other professors refer to them as "Euler Diagrams."
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