Rule of Material Equivalence/Formulation 1
Theorem
The biconditional operation can be interpreted as the conjunction of conditionals:
- $p \iff q \dashv \vdash \paren {p \implies q} \land \paren {q \implies p}$
Proof 1
By the tableau method of natural deduction:
| Line | Pool | Formula | Rule | Depends upon | Notes | |
|---|---|---|---|---|---|---|
| 1 | 1 | $p \iff q$ | Premise | (None) | ||
| 2 | 1 | $p \implies q$ | Biconditional Elimination: $\iff \EE_1$ | 1 | ||
| 3 | 1 | $q \implies p$ | Biconditional Elimination: $\iff \EE_2$ | 1 | ||
| 4 | 1 | $\paren {p \implies q} \land \paren {q \implies p}$ | Rule of Conjunction: $\land \II$ | 2, 3 |
By the tableau method of natural deduction:
| Line | Pool | Formula | Rule | Depends upon | Notes | |
|---|---|---|---|---|---|---|
| 1 | 1 | $\paren {p \implies q} \land \paren {q \implies p}$ | Premise | (None) | ||
| 2 | 1 | $p \implies q$ | Rule of Simplification: $\land \EE_1$ | 1 | ||
| 3 | 1 | $q \implies p$ | Rule of Simplification: $\land \EE_2$ | 1 | ||
| 4 | 1 | $p \iff q$ | Biconditional Introduction: $\iff \II$ | 2, 3 |
$\blacksquare$
Proof by Truth Table
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.
$\begin{array}{|ccc|ccccccc|} \hline p & \iff & q & (p & \implies & q) & \land & (q & \implies & p) \\ \hline \F & \T & \F & \F & \T & \F & \T & \F & \T & \F \\ \F & \F & \T & \F & \T & \T & \F & \T & \F & \F \\ \T & \F & \F & \T & \F & \F & \F & \F & \T & \T \\ \T & \T & \T & \T & \T & \T & \T & \T & \T & \T \\ \hline \end{array}$
$\blacksquare$
Sources
- 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $1$: The Propositional Calculus $1$: $4$ The Biconditional
- 1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.1$: What is a Set?
- 2008: David Joyner: Adventures in Group Theory (2nd ed.) ... (previous) ... (next): Chapter $1$: Elementary, my dear Watson: $\S 1.1$: You have a logical mind if...: Ponderable $1.1.1$
- 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): $\S 2.3.3$