Use a truth table to prove \(\sim ( \sim \mathrm { T } \Rightarrow \sim \mathrm { S } ) \Leftrightarrow ( \sim \mathrm { T } \wedge \mathrm { S } )\).
Prove that \(( \mathrm { A } \Rightarrow \mathrm { B } ) \Leftrightarrow ( \sim \mathrm { A } \vee \mathrm { B } )\) and hence use Boolean algebra to prove that
$$\sim ( \sim \mathrm { T } \Rightarrow \sim \mathrm {~S} ) \Leftrightarrow ( \sim \mathrm { T } \wedge \mathrm {~S} ) .$$
A teacher wrote on a report "It is not the case that if Joanna doesn't try then she won't succeed." He meant to say that if Joanna were to try then she would have a chance of success. By letting T be "Joanna will try" and S be "Joanna will succeed", find the real meaning of what the teacher wrote.