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Math Help - Construct a truth table for the compound proposition

  1. #1
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    Construct a truth table for the compound proposition

    I need to make a truth table for this compound proposition:

    (q →p) ↔ (p ↔ q)

    When looking at examples I noticed that between p and q, p always comes first in the truth table. So do I need to switch around the proposition to say:

    (p ↔ q) ↔ (q →p) ?

    I feel like if I leave it as is originally then instead of putting the table like this:

    p q
    T T
    T F
    F T
    F F


    I would have to do this:

    q p
    T T
    T F
    F T
    F F



    Am I over-thinking the problem or am I on the right track? Someone please provide some assistance. I am so lost.
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  2. #2
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    Re: Construct a truth table for the compound proposition

    Quote Originally Posted by lamentofking View Post
    Am I over-thinking the problem
    Yes. In a truth table, variable names (i.e., names of columns) are usually listed in the alphabetical order, but this is not a law. In a formula, variables can occur in an arbitrary order. Formulas are just given to you, and you are not supposed to change them just to construct a truth table. There is nothing wrong in having a truth table with two columns p and q and the third column listing the truth values of the formula (q →p) ↔ (p ↔ q). This is just like having the following table listing values of an arithmetic expression.
    Code:
    x y | y * x
    1 2 |   2
    3 5 |   15
    4 2 |   8
    3 3 |   9
    Here there is no need to switch the x and y columns even though the first variable in (y * x) is y.

    You could also make the q column first. It would still be a valid truth table, but the order of truth values in the third column would be different, which may make it more difficult for your instructor to check your work.
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  3. #3
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    Re: Construct a truth table for the compound proposition

    Hello, lamentofking!

    Your question indicates that this is your first truth table.


    \text{Construct a truth table for: }\:(q \to\,\sim\!p)\:\longleftrightarrow\:(p \leftrightarrow q)

    \begin{array}{|c|c|ccccccc|} p & q & (q & \to & \sim\!p) & \longleftrightarrow & (p & \leftrightarrow & q) \\ \hline T&T& T&F&F & \color{blue}{F} &T&T&T \\ T&F & F&T&F & \color{blue}{F} & T&F&F \\ F&T & T&T&T & \color{blue}{F} & F&F&T \\ F&F & F&T&T & \color{blue}{T} & F&T&F \\ \hline &&1&2&1 &3& 1&2&1 \\ \hline\end{array}


    The numbers indicate the order in which the columns are filled.
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