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Math Help - Truth Tables and Euler Circles

  1. #1
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    Question Truth Tables and Euler Circles

    I am having a hard time understanding truth tables. Anyone able to show me a truth table for the following 2 problems?

    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


    help with these will be greatly appreciated I am having major issues understanding truth tables and my work is overdue.
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  2. #2
    Super Member

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    Lexington, MA (USA)
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    Hello, Maddog!

    There seem to be two major difficulties:
    . . [1] Translating statements into logic
    . . [2] Constructing and completing truth tables.
    Can you describe yours?


    1) .No A is B . . . . . . . {\color{blue}a \to \sim b}
    . . .Some C is A . . . . . {\color{blue}c \wedge a}
    . . .------------------- . ---------
    . \therefore Some C is not B . . {\color{blue} c \:\wedge \sim b}

    \begin{array}{|c|c|c||ccccccccccc|}<br />
a & b & c & [(a & \to & \sim b) & \wedge & (c & \wedge & a)] & \to & (c & \wedge & \sim b) \\ \hline \hline<br />
T&T&T & T&F&F &F& T&T&T & T & T&F&F \\<br />
T&T&F & T&F&F &F& F&F&T & T & F&F&F \\<br />
T&F&T & T&T&T &T& T&T&T &T& T&T&T \\<br />
T&F&F & T&T&T &F& F&F&T &T& F&F&T \\<br />
F&T&T & F&T&F &F& T&F&F &T& T&F&F \\<br />
F&T&F& F&T&F &F& F&F&F &T& F&F&F \\<br />
F&F&T & F&T&T &F& T&F&F &T& T&T&T  \end{array}
    \begin{array}{|c|c|c||ccccccccccc|}F&F&F & F&\;T&\;\;\;T &\;\;F& F&F&F &\;\;T& F&\;T&\;\;T\;\; \\ \hline \hline<br />
& & & 1 & 2 & 1 & 3 & 1 & 2 & 1 & 4 & 1 & 2 & 1 \end{array}

    The last column (4) is all T's.
    The statement is valid.


    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~


    Now try the second problem . . .


    2) All B is A . . {\color{blue} b \to a}
    . . All C is A . . {\color{blue}c \to a}
    . . ----------- . .-------
    . \therefore All C is B . . {\color{blue}c \to b}

    Now test: . \bigg[(b \to a) \wedge (c \to a)\bigg] \to (c \to b)

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