All following laws are #logically equivalence. t denotes tautology (always true) and c denotes contradiction (always false). \(\equiv\) indicate that the left-hand side is identical to the right-hand side.
Compound Statement
Following shows the laws applied on #202205061208:
Commutative laws: \(p \land q \equiv q \land p\) and \(p \lor q \equiv q \lor p\)
Associative laws: \((p \land q) \land r \equiv p \land (q \land r)\) and \((p \lor q) \lor r \equiv p \lor (q \lor r)\)
Distributive laws: \(p \land (q \lor r) \equiv (p \land q) \lor (p \land r)\) and \(p \lor (q \land r) \equiv (p \lor q) \land (p \lor r)\)
Identity laws: \(p \land \textbf{t} \equiv p\) and \(p \lor \textbf{c} \equiv p\)
Negation laws: \(p \lor \sim p \equiv \textbf{t}\) and \(p \land \sim p \equiv \textbf{c}\)
Double negative law: \(\sim (\sim p) \equiv p\)
Idempotent laws: \(p \land p \equiv p\) and \(p \lor p \equiv p\)
Universal bound laws: \(p \lor \textbf{t} \equiv \textbf{t}\) and \(p \land \textbf{c} \equiv \textbf{c}\)
De Morgan’s laws: \(\sim (p \land q) \equiv \sim p \lor \sim q\) and \(\sim (p \lor q) \equiv \sim p \land \sim q\)
Absorption laws: \(p \lor (p \land q) \equiv p\) and \(p \land (p \lor q) \equiv p\)
Negations of t and c: \(\sim \textbf{t} \equiv \textbf{c}\) and \(\sim \textbf{c} \equiv \textbf{t}\)
Conditional Statement
Following shows the laws applied on #202205062055:
Conditional to Compound: \(p \rightarrow q \equiv \sim p \lor q\)
\(p \lor q \rightarrow r \equiv (p \rightarrow r) \land (q \rightarrow r)\)
Negation: \(\sim (p \rightarrow q) \equiv p \land \sim q\)
Contrapositive: \(p \rightarrow q \equiv \sim q \rightarrow \sim p\)
Converse and Inverse: \(q \rightarrow p \equiv \sim p \rightarrow \sim q\)
Bidirectional: \(p \leftrightarrow q \equiv (\sim p \lor q) \land (\sim q \lor p)\)