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Thread: Linked iff statements

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    Forum Admin topsquark's Avatar
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    Linked iff statements

    Sorry if this is overly simple but I've never actually done a logic course beyond truth tables. I can guess the meaning but I've never encountered this one.

    If I have the statement $\displaystyle X \Leftrightarrow Y \Leftrightarrow Z$ is this just a shorthand way of writing the three statements $\displaystyle X \Leftrightarrow Y$, $\displaystyle Y \Leftrightarrow Z$, and $\displaystyle X \Leftrightarrow Z$?

    I can give you the actual problem statement if you need it.

    Thanks!

    -Dan
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    Re: Linked iff statements

    Quote Originally Posted by topsquark View Post
    Sorry if this is overly simple but I've never actually done a logic course beyond truth tables. I can guess the meaning but I've never encountered this one. If I have the statement $\displaystyle X \Leftrightarrow Y \Leftrightarrow Z$ is this just a shorthand way of writing the three statements $\displaystyle X \Leftrightarrow Y$, $\displaystyle Y \Leftrightarrow Z$, and $\displaystyle X \Leftrightarrow Z$?
    The statement that $X\iff Y$ means that $X$ is true if and only if $Y$ is true.
    That is in a truth table the truth values are the same: SEE HERE
    Please note the last line $F~F~\to T$ that is having the same truth value is true.
    Thus then $\iff$ is transitive.
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    Forum Admin topsquark's Avatar
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    Re: Linked iff statements

    Quote Originally Posted by Plato View Post
    The statement that $X\iff Y$ means that $X$ is true if and only if $Y$ is true.
    That is in a truth table the truth values are the same: SEE HERE
    Please note the last line $F~F~\to T$ that is having the same truth value is true.
    Thus then $\iff$ is transitive.
    Okay, thanks. Just checking.

    -Dan
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    Re: Linked iff statements

    Linked iff statements-capture1.jpg

    I tried using a truth table to prove the transitive property. Does the implication in column 8 show that?

    I'm convinced it does but I'd just like a second eye. This table shows that for all possible truth values of X, Y, and Z, the statement if (X iff Y)^(Y iff Z) then (X iff Z) is a tautology -- that is (X iff Y)^(Y iff Z) is equivalent to (X iff Z).
    Last edited by Elusive1324; Jul 1st 2019 at 08:10 AM.
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