Question

The logically equivalent statement of \[\left(\sim p\wedge q\right)\vee\left(\sim p\wedge\sim q\right)\] \[\vee\left(\ p\wedge\sim q\right)\] is

Options

• \[(\sim p)\wedge q\]

• \[(\sim p)\lor(\sim q)\]

• \[(\sim p)\wedge(\sim q)\]

• \[\ p\lor\ q\]

Answer 1

\[(\sim p)\lor(\sim q)\]

Explanation:

\[(\sim\mathrm{p}\wedge\mathrm{q})\vee(\sim\mathrm{p}\wedge\sim\mathrm{q})\vee(\mathrm{p}\wedge\sim\mathrm{q})\]

\[\equiv\sim\mathrm{p}\wedge(\mathrm{q}\vee\sim\mathrm{q})\vee(\mathrm{p}\wedge\sim\mathrm{q})\]

\[\equiv({\sim}\mathrm{p}\wedge\mathrm{T})\vee(\mathrm{p}\wedge{\sim}\mathrm{q})\]

\[\equiv\sim\mathrm{p}\lor(\mathrm{p}\land\sim\mathrm{q})\]

\[\equiv(\sim\mathrm{p}\lor\mathrm{p})\land(\sim\mathrm{p}\lor\sim\mathrm{q})\]

\[\equiv\mathrm{T}\wedge(\sim\mathrm{p}\vee\sim\mathrm{q})\]

\[\equiv\sim\mathrm{p}\lor\sim\mathrm{q}\]