Rule of Association/Conjunction/Formulation 1
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Theorem
- $p \land \left({q \land r}\right) \dashv \vdash \left({p \land q}\right) \land r$
Proof 1
By the tableau method of natural deduction:
| Line | Pool | Formula | Rule | Depends upon | Notes | |
|---|---|---|---|---|---|---|
| 1 | 1 | $p \land \left({q \land r}\right)$ | Premise | (None) | ||
| 2 | 1 | $p$ | Rule of Simplification: $\land \EE_1$ | 1 | ||
| 3 | 1 | $q \land r$ | Rule of Simplification: $\land \EE_2$ | 1 | ||
| 4 | 1 | $q$ | Rule of Simplification: $\land \EE_1$ | 3 | ||
| 5 | 1 | $r$ | Rule of Simplification: $\land \EE_2$ | 3 | ||
| 6 | 1 | $p \land q$ | Rule of Conjunction: $\land \II$ | 2, 4 | ||
| 7 | 1 | $\left({p \land q}\right) \land r$ | Rule of Conjunction: $\land \II$ | 6, 5 |
By the tableau method of natural deduction:
| Line | Pool | Formula | Rule | Depends upon | Notes | |
|---|---|---|---|---|---|---|
| 1 | 1 | $\left({p \land q}\right) \land r$ | Premise | (None) | ||
| 2 | 1 | $p \land q$ | Rule of Simplification: $\land \EE_1$ | 1 | ||
| 3 | 1 | $r$ | Rule of Simplification: $\land \EE_2$ | 1 | ||
| 4 | 1 | $p$ | Rule of Simplification: $\land \EE_1$ | 2 | ||
| 5 | 1 | $q$ | Rule of Simplification: $\land \EE_2$ | 2 | ||
| 6 | 1 | $q \land r$ | Rule of Conjunction: $\land \II$ | 5, 3 | ||
| 7 | 1 | $p \land \left({q \land r}\right)$ | Rule of Conjunction: $\land \II$ | 4, 6 |
$\blacksquare$
Proof 2
We apply the Method of Truth Tables.
As can be seen by inspection, the truth values under the main connectives match for all boolean interpretations.
$\begin{array}{|ccccc||ccccc|} \hline
p & \land & (q & \land & r) & (p & \land & q) & \land & r \\
\hline
F & F & F & F & F & F & F & F & F & F \\
F & F & F & F & T & F & F & F & F & T \\
F & F & T & F & F & F & F & T & F & F \\
F & F & T & T & T & F & F & T & F & T \\
T & F & F & F & F & T & F & F & F & F \\
T & F & F & F & T & T & F & F & F & T \\
T & F & T & F & F & T & T & T & F & F \\
T & T & T & T & T & T & T & T & T & T \\
\hline
\end{array}$
$\blacksquare$
Sources
- 2000: Michael R.A. Huth and Mark D. Ryan: Logic in Computer Science: Modelling and reasoning about systems ... (previous) ... (next): $\S 1.2.1$: Rules for natural deduction: Exercise $1.2: \ 3$
- 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): $\S 2.3.3$