[Math] Is it possible for one of $(\phi\wedge\psi)\wedge\theta$ and $\phi\wedge(\psi\wedge\theta)$ to be true and the other false

logicproof-verificationproof-writing

I'm taking 'Introduction to Mathematical Thinking' and is asked :

Is it possible for one of $(\phi\wedge\psi)\wedge\theta$ and $\phi\wedge(\psi\wedge\theta)$ to be true and the other false ? (If not then the associative property holds for conjunction) Reading https://en.wikipedia.org/wiki/Associative_property the associative property for conjunction holds :

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Here is my solution :

Conjunction is true of both conjunctions are true.
$$\begin{array} {|c|}
\hline
\phi & \psi & \theta & \phi\wedge\psi & (\phi\wedge\psi)\wedge\theta & \phi\wedge(\psi\wedge\theta) & && \\ \hline
T& T& T& T& T& T& \\ \hline
T& F& T& F& F& F& \\ \hline
T& T& F& T& F& T& \\ \hline
T& T& F& T& F& T& \\ \hline
T& F& F& F& F& F& \\ \hline
F& F& T& F& F& F& \\ \hline
F& T& F& F& F& F& \\ \hline
F& F& F& F& F& F& \\ \hline
\end{array}$$

From this truth $(\phi\wedge\psi)\wedge\theta$ $\nrightarrow$ $\phi\wedge(\psi\wedge\theta)$ and $(\phi\wedge\psi)\wedge\theta$ $\nleftarrow$ $\phi\wedge(\psi\wedge\theta)$ so answer is yes : It is possible for one of $(\phi\wedge\psi)\wedge\theta$ and $\phi\wedge(\psi\wedge\theta)$ to be true and the other false. But I believe this to be incorrect , is there a mistake in my truth table ?

Best Answer

There are a few mistakes in your truth table. The third and fourth rows have the same values for $\phi$, $\varphi$ and $\theta$ so you have missed out one combination. And some of your truth values for $\phi\wedge(\varphi\wedge\theta)$ are wrong. It might be easier to work these out if you have a column for $\varphi\wedge\theta$, as you did for $\phi\wedge\varphi$ when working out the values for the first conjunction.

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