Tag: proofs in mathematics

$\left( {p \to q} \right) \cong \left( { \sim q \to \sim p} \right)$

$ \sim \left( {p \vee q} \right) \cong \left( { \sim p \vee \sim q} \right)$

$ \sim \left( {p \to q} \right) \cong \left( {p \vee \sim q} \right)$

$ \sim \left( {p \wedge q} \right) \cong \left( { \sim p \wedge \sim q} \right)$

$p$

$q$

$ \sim q $

$ \sim p $

True

False

$q$

$p\vee q$

$p$

$p\vee \sim q$

$p \vee \sim q$

$p \wedge \sim q$

$ \sim p \vee q$

$ \sim p \wedge q$

True

False

$\left(p \rightarrow q\right)\leftrightarrow \left( \sim q \rightarrow \sim p \right) $

$\left[ \left( p\rightarrow q \right) \wedge \left( q\rightarrow r \right) \right]\rightarrow \left( p\rightarrow r \right) $

$\left( \sim p\vee q \right) \leftrightarrow \left( p\wedge \sim q \right) $

$p \rightarrow \sim p$

$p \vee ( q \wedge r )$

$( p \wedge q ) \wedge r$

$( p \vee q ) \vee r$

$p \rightarrow ( q \wedge r )$

$p \text { is true and } q \text{ is true}$

$p \text { is false and } q  \text{ is true}$

$p \text { is true and } q \text{ is false}$

both $p$ and $q$ are false

Mathematics is interesting implies that Mathematics is difficult

Mathematics is interesting implies and is implied by Mathematics is difficult

Mathematics is interesting and Mathematics is difficult

Mathematics is interesting or Mathematics is difficult