Difference between compound statements using symbols

[aburchett]

What is the difference between ~p?~q and ~(p?q) when reading these two statements?

 

[soroban]

Hello, aburchett!

\(\displaystyle \text{What is the difference between }\,\sim p\; \wedge \sim q\,\text{ and }\,\sim(p \,\wedge\, q)\,\text{ when reading these two statements?}\)

\(\displaystyle \sim p \:\wedge\:\sim q\,\text{ says: }p\text{ is false and }q\text{ is false.}\)

. . That is, both statements are false.


\(\displaystyle \sim(p\,\wedge\,q)\,\text{ says: It is not true that both }p\text{ and }q\text{ are true.}\)

. . That is, at least one of the statements is false.
. . . \(\displaystyle (p\text{ is false, or }q\text{ is false, or both are false.)}\)