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Show that the set of connectives $\{\wedge, \leftrightarrow, \oplus\}$ is adequate, where $\oplus$ is defined by the truth table:

$\begin{array}{|c | c | c |} \hline p & q & p \oplus q \\ \hline 1 & 1 & 0 \\ \hline 1 & 0 & 1 \\ \hline 0 & 1 & 1 \\ \hline 0 & 0 & 0 \\ \hline \end{array}$

I've been trying all day and for the life of me I can't get my head around this concept of proving a set of connectives to be adequate. I've read every written answer on stackexchange and nothing quite explains it. If I could get some help on this I would truly appreciate it.

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  • $\begingroup$ It's hard to know what to say because you don't say what your difficulty is or what you don't understand. Do you know the definition of "adequate set of connectives"? If not, have you looked it up? If you did, what part isn't clear? $\endgroup$
    – MJD
    Commented Feb 10, 2015 at 13:51

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Note that you can write $\neg p$ as $p\leftrightarrow (p\oplus p)$. Since $\{\neg,\wedge\}$ is adequate, so $\{\wedge,\leftrightarrow,\oplus\}$ is.

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