# Rule of Simplification/Sequent Form/Formulation 2

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## Theorem

The Rule of Simplification can be symbolised by the sequents:

- $(1): \quad \vdash p \land q \implies p$
- $(2): \quad \vdash p \land q \implies q$

### Form 1

- $\vdash p \land q \implies p$

### Form 2

- $\vdash p \land q \implies q$

## Proof 1

### Form 1

By the tableau method of natural deduction:

Line | Pool | Formula | Rule | Depends upon | Notes | |
---|---|---|---|---|---|---|

1 | 1 | $p \land q$ | Assumption | (None) | ||

2 | 1 | $p$ | Rule of Simplification: $\land \mathcal E_1$ | 1 | ||

3 | $p \land q \implies p$ | Rule of Implication: $\implies \mathcal I$ | 1 – 2 | Assumption 1 has been discharged |

$\blacksquare$

### Form 2

By the tableau method of natural deduction:

Line | Pool | Formula | Rule | Depends upon | Notes | |
---|---|---|---|---|---|---|

1 | 1 | $p \land q$ | Assumption | (None) | ||

2 | 1 | $q$ | Rule of Simplification: $\land \mathcal E_2$ | 1 | ||

3 | $p \land q \implies q$ | Rule of Implication: $\implies \mathcal I$ | 1 – 2 | Assumption 1 has been discharged |

$\blacksquare$

## Proof 2

We apply the Method of Truth Tables.

As can be seen by inspection, the truth values under the main connective are $T$ for all boolean interpretations.

$\begin{array}{|ccc|c|c||c|c|} \hline p & \land & q & p & q & p \land q \implies p & p \land q \implies q \\ \hline F & F & F & F & F & T & T \\ F & F & T & F & T & T & T \\ T & F & F & T & F & T & T \\ T & T & T & T & T & T & T \\ \hline \end{array}$

$\blacksquare$