Math Problem Solver

Logic Expression
Symbol: $\left(\left(p \rightarrow q\right) \vee \left(\neg q \wedge r\right)\right) \leftrightarrow \left(r \rightarrow q\right)$
Text: $\left(\left(p \text{ implies } q\right) \text{ or } \left(\text{not } q \text{ and } r\right)\right) \text{ if and only if } \left(r \text{ implies } q\right)$

Classification contingency

Truth Table

Legend

Simplification$$\begin{gathered} \left(\left(p \rightarrow q\right) \vee \left(\neg q \wedge r\right)\right) \leftrightarrow \left(r \rightarrow q\right) & \equiv & \left(\neg p \vee q \vee \left(\neg q \wedge r\right)\right) \leftrightarrow \left(r \rightarrow q\right) & \text{Conditional Equivalence} \\ & \equiv & \left(\neg p \vee q \vee r\right) \leftrightarrow \left(r \rightarrow q\right) & \text{Redundancy Law (2)} \\ & \equiv & \left(\neg p \vee q \vee r\right) \leftrightarrow \left(\neg r \vee q\right) & \text{Conditional Equivalence} \\ & \equiv & \left(\neg p \vee q \vee r \vee \neg \left(\neg r \vee q\right)\right) \wedge \left(\neg \left(\neg p \vee q \vee r\right) \vee \neg r \vee q\right) & \text{Biconditional Equivalence} \\ & \equiv & \left(\neg p \vee q \vee r \vee \left(\neg \left(\neg r\right) \wedge \neg q\right)\right) \wedge \left(\neg \left(\neg p \vee q \vee r\right) \vee \neg r \vee q\right) & \text{De Morgan's Law} \\ & \equiv & \left(\neg p \vee q \vee r \vee \left(r \wedge \neg q\right)\right) \wedge \left(\neg \left(\neg p \vee q \vee r\right) \vee \neg r \vee q\right) & \text{Double Negation Law} \\ & \equiv & \left(\neg p \vee q \vee r\right) \wedge \left(\neg \left(\neg p \vee q \vee r\right) \vee \neg r \vee q\right) & \text{Absorption Law} \\ & \equiv & \left(\neg p \vee q \vee r\right) \wedge \left(\neg r \vee q\right) & \text{Redundancy Law (2)} \\ & \equiv & q \vee \left(\left(\neg p \vee r\right) \wedge \neg r\right) & \text{Distributive Law} \\ & \equiv & q \vee \left(\neg p \wedge \neg r\right) & \text{Redundancy Law (2)} \end{gathered}$$ Note: Solution may not be as simplified as possible.

Conjunctive Normal Form$$\begin{gathered} q \vee \left(\neg p \wedge \neg r\right) & \equiv & \left(q \vee \neg p\right) \wedge \left(q \vee \neg r\right) & \text{Distributive Law} \end{gathered}$$ Note: Solution may not be as simplified as possible.

Note 1: These equivalences and tautologies are used to generate the above steps.
Note 2: Two logical statements are logically equivalent if they always produce the same truth value. Consequently, p ≡ q is same as saying p ↔ q is a tautology. $$\begin{array}{c|c}\textbf{Equivalence} \\ \hline \text{Absorption Law} & \begin{gathered} p \wedge \left(p \vee q\right) \equiv p \\ p \vee \left(p \wedge q\right) \equiv p \end{gathered} \\ \hline \text{Biconditional Equivalence} & \begin{gathered} p \leftrightarrow q \equiv \left(p \vee \neg q\right) \wedge \left(\neg p \vee q\right) \\ p \leftrightarrow q \equiv \left(p \wedge q\right) \vee \left(\neg p \wedge \neg q\right) \end{gathered} \\ \hline \text{Biconditional Simplification} & \begin{gathered} p \leftrightarrow p \equiv \text{True} & p \leftrightarrow \text{True} \equiv p \\ p \leftrightarrow \neg p \equiv \text{False} & p \leftrightarrow \text{False} \equiv \neg p \end{gathered} \\ \hline \text{Complement Law} & \begin{gathered} p \wedge \neg p \equiv \text{False} \\ p \vee \neg p \equiv \text{True} \end{gathered} \\ \hline \text{Conditional Equivalence} & p \rightarrow q \equiv \neg p \vee q \\ \hline \text{Conditional Simplification} & \begin{gathered} p \rightarrow p \equiv \text{True} & p \rightarrow \text{True} \equiv \text{True} & p \rightarrow \text{False} \equiv \neg p \\ p \rightarrow \neg p \equiv \neg p & \text{True} \rightarrow p \equiv p & \text{False} \rightarrow p \equiv \text{True} \\ \neg p \rightarrow p \equiv p\end{gathered} \\ \hline \text{Consensus Law} & \begin{gathered} \left(p \vee q\right) \wedge \left(\neg p \vee r\right) \wedge \left(q \vee r\right) \equiv \left(p \vee q\right) \wedge \left(\neg p \vee r\right) \\ \left(p \wedge q\right) \vee \left(\neg p \wedge r\right) \vee \left(q \wedge r\right) \equiv \left(p \wedge q\right) \vee \left(\neg p \wedge r\right) \end{gathered} \\ \hline \text{De Morgan's Law} & \begin{gathered} \neg \left(p \wedge q\right) \equiv \neg p \vee \neg q \\ \neg \left(p \vee q\right) \equiv \neg p \wedge \neg q \end{gathered} \\ \hline \text{Distributive Law} & \begin{gathered} p \wedge \left(q \vee r\right) \equiv \left(p \wedge q\right) \vee \left(p \wedge r\right) \\ p \vee \left(q \wedge r\right) \equiv \left(p \vee q\right) \wedge \left(p \vee r\right) \\ \left(p \vee q\right) \wedge \left(r \vee s\right) \equiv \left(p \wedge r\right) \vee \left(p \wedge s\right) \vee \left(q \wedge r\right) \vee \left(q \wedge s\right) \\ \left(p \wedge q\right) \vee \left(r \wedge s\right) \equiv \left(p \vee r\right) \wedge \left(p \vee s\right) \wedge \left(q \vee r\right) \wedge \left(q \vee s\right) \end{gathered} \\ \hline \text{Domination Law} & \begin{gathered} p \vee \text{True} \equiv \text{True} \\ p \wedge \text{False} \equiv \text{False} \end{gathered} \\ \hline \text{Double Negation Law} & \neg \left(\neg p\right) \equiv p \\ \hline \text{Idempotent Law} & \begin{gathered} p \wedge p \equiv p \\ p \vee p \equiv p \end{gathered} \\ \hline \text{Identity Law} & \begin{gathered} p \wedge \text{True} \equiv p \\ p \vee \text{False} \equiv p \end{gathered} \\ \hline \text{NAND} & p \uparrow q \equiv \neg \left(p \wedge q\right) \\ \hline \text{Negation Law} & \begin{gathered} \neg \text{True} \equiv \text{False} \\ \neg \text{False} \equiv \text{True} \end{gathered} \\ \hline \text{NOR} & p \downarrow q \equiv \neg \left(p \vee q\right) \\ \hline \text{Negation of Biconditional Equivalence} & \begin{gathered} \neg \left(p \leftrightarrow q\right) \equiv \left(p \vee q\right) \wedge \left(\neg p \vee \neg q\right) \\ \neg \left(p \leftrightarrow q\right) \equiv \left(p \wedge \neg q\right) \vee \left(\neg p \wedge q\right) \end{gathered} \\ \hline \text{Negation of Conditional Equivalence} & \neg \left(p \rightarrow q\right) \equiv p \wedge \neg q \\ \hline \text{Redundancy Law (1)} & \begin{gathered} \left(p \vee q\right) \wedge \left(p \vee \neg q\right) \equiv p \\ \left(p \wedge q\right) \vee \left(p \wedge \neg q\right) \equiv p \end{gathered} \\ \hline \text{Redundancy Law (2)} & \begin{gathered} p \wedge \left(\neg p \vee q\right) \equiv p \wedge q \\ p \vee \left(\neg p \wedge q\right) \equiv p \vee q \end{gathered} \\ \hline \text{XOR} & \begin{gathered} p \oplus q \equiv \left(p \vee q\right) \wedge \left(\neg p \vee \neg q\right) \\ p \oplus q \equiv \left(p \wedge \neg q\right) \vee \left(\neg p \wedge q\right) \end{gathered} \\ \hline \text{XOR Simplification} & \begin{gathered} p \oplus p \equiv \text{False} & p \oplus \text{True} \equiv \neg p \\ p \oplus \neg p \equiv \text{True} & p \oplus \text{False} \equiv p \end{gathered} \\ \hline \text{XNOR} & p \odot q \equiv \neg \left(p \oplus q\right) \end{array}$$ $$\begin{array}{c|c}\textbf{Tautology} \\ \hline \text{Conjunctive Simplification} & \begin{gathered} \left(p \wedge q\right) \rightarrow p \\ \left(p \wedge q\right) \rightarrow q \end{gathered} \\ \hline \text{Contradiction} & \neg \left(p \wedge \neg p\right) \\ \hline \text{Contrapositive} & \left(p \rightarrow q\right) \leftrightarrow \left(\neg q \rightarrow \neg p\right) \\ \hline \text{Disjunctive Addition} & \begin{gathered} p \rightarrow \left(p \vee q\right) \\ q \rightarrow \left(p \vee q\right) \end{gathered} \\ \hline \text{Disjunctive Syllogism} & \begin{gathered} \left(\left(p \vee q\right) \wedge \neg q\right) \rightarrow p \\ \left(\left(p \vee q\right) \wedge \neg p\right) \rightarrow q \end{gathered} \\ \hline \text{Hypothetical Syllogism} & \left(\left(p \rightarrow q\right) \wedge \left(q \rightarrow r\right)\right) \rightarrow \left(p \rightarrow r\right) \\ \hline \text{Modus Ponens} & \left(p \wedge \left(p \rightarrow q\right)\right) \rightarrow q \\ \hline \text{Modus Tollens} & \left(\neg q \wedge \left(p \rightarrow q\right)\right) \rightarrow \neg p \end{array}$$