Rule of Simplification/Sequent Form/Formulation 2/Proof 2
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Theorem
- $(1): \quad \vdash p \land q \implies p$
- $(2): \quad \vdash p \land q \implies q$
Proof
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.
$\begin{array}{|ccc|c|c||c|c|} \hline p & \land & q & p & q & p \land q \implies p & p \land q \implies q \\ \hline F & F & F & F & F & T & T \\ F & F & T & F & T & T & T \\ T & F & F & T & F & T & T \\ T & T & T & T & T & T & T \\ \hline \end{array}$
$\blacksquare$
Sources
- 1946: Alfred Tarski: Introduction to Logic and to the Methodology of Deductive Sciences (2nd ed.) ... (previous) ... (next): $\S \text{II}.13$: Symbolism of sentential calculus
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): truth table
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): truth table