Formulae
$\gdef \N{\mathbb{N}} \gdef \Z{\mathbb{Z}} \gdef \Q{\mathbb{Q}} \gdef \R{\mathbb{R}} \gdef \C{\mathbb{C}} \gdef \setcomp#1{\overline{#1}} \gdef \sseq{\subseteq} \gdef \pset#1{\mathcal{P}(#1)} \gdef \covariant{\operatorname{Cov}} \gdef \of{\circ} \gdef \p{^{\prime}} \gdef \pp{^{\prime\prime}} \gdef \ppp{^{\prime\prime\prime}} \gdef \pn#1{^{\prime\times{#1}}} $
Sentential Logic
Equivalences
| Name |
Formula |
Notes |
| E1 |
\(R \lor \Bbb T \iff \Bbb T\) |
|
| E2 |
\(R \lor \Bbb F \iff R\) |
|
| E3 |
\(R \land \Bbb F \iff \Bbb F\) |
|
| E4 |
\(R \land \Bbb T \iff R\) |
|
| E5 |
\(R \lor R \iff R\) |
Idempotent Law |
| E6 |
\(R \land R \iff R\) |
Idempotent Law |
| E7 |
\(R \lor (\neg R) \iff \Bbb T\) |
Tautology |
| E8 |
\(R \land (\neg R) \iff \Bbb F\) |
Contradiction |
| E9 |
\(\neg (\neg R) \iff R\) |
Double Negation Law |
| E10 |
\(R \lor S \iff S \lor R\) |
Commutative Law |
| E11 |
\(R \land S \iff S \land R\) |
Commutative Law |
| E12 |
\(R \lor (S \lor Q) \iff (R \lor S) \lor Q\) |
Associative Law |
| E13 |
\(R \land (S \land Q) \iff (R \land S) \land Q\) |
Associative Law |
| E14 |
\(R \lor (S \land Q) \iff (R \lor S) \land (R \lor Q)\) |
Distributive Law |
| E15 |
\(R \land (S \lor Q) \iff (R \land S) \lor (R \land Q)\) |
Distributive Law |
| E16 |
\(\neg (R \lor S) \iff (\neg R) \land (\neg S)\) |
De Morgan’s Law |
| E17 |
\(\neg (R \land S) \iff (\neg R) \lor (\neg S)\) |
De Morgan’s Law |
| E18 |
\(R \lor (R \land S) \iff R\) |
Absorption Law |
| E19 |
\(R \land (R \lor S) \iff R\) |
Absorption Law |
| E20 |
\(R \rightarrow S \iff (\neg R) \lor S\) |
Implication Law |
| E21 |
\(R \rightarrow S \iff (\neg S) \rightarrow (\neg R)\) |
Contrapositive Law |
| E22 |
\(R \leftrightarrow S \iff (R \rightarrow S) \land (S \rightarrow R)\) |
Biconditional Law |
| E23 |
\(R \rightarrow (S \rightarrow Q) \iff (R \land S) \rightarrow Q\) |
Exportation Law |