Disjunction with Tautology

Theorem

A disjunction with a tautology:

$p \lor \top \dashv \vdash \top$

Proof by Natural Deduction

By the tableau method of natural deduction:

$p \lor \top \vdash \top$
Line Pool Formula Rule Depends upon Notes
1 1 $p \lor \top$ Premise (None)
2 2 $\top$ Assumption (None)
3 3 $p$ Assumption (None)
4 3 $p \lor \neg p$ Rule of Addition: $\lor \II_1$ 3
5 3 $\top$ Law of Excluded Middle 4
6 1 $\top$ Proof by Cases: $\text{PBC}$ 1, 2 – 2, 3 – 5 Assumptions 2 and 3 have been discharged

$\Box$

By the tableau method of natural deduction:

$\top \vdash p \lor \top$
Line Pool Formula Rule Depends upon Notes
1 1 $\top$ Premise (None)
2 1 $p \lor \top$ Rule of Addition: $\lor \II_2$ 1

$\blacksquare$

Proof by Truth Table

We apply the Method of Truth Tables to the proposition.

As can be seen by inspection, in each case, the truth values in the appropriate columns match for all boolean interpretations.

$\begin{array}{|c|c||ccc|} \hline p & \top & p & \lor & \top \\ \hline F & T & F & T & T \\ T & T & T & T & T \\ \hline \end{array}$

$\blacksquare$