Mickey ate the last of the cheese. Minnie was put out at this. Mickey's defence was "There wasn't enough left not to eat it all".
Let "c" represent "there is enough cheese for two" and "e" represent "one person can eat all of the cheese".
Which of the following best captures Mickey's argument?
$$\mathrm { c } \Rightarrow \mathrm { e } \quad \mathrm { c } \Rightarrow \sim \mathrm { e } \quad \sim _ { \mathrm { c } } \Rightarrow \mathrm { e } \quad \sim _ { \mathrm { c } } \Rightarrow \sim \mathrm { e }$$
In the ensuing argument Minnie concedes that if there's a lot left then one should not eat it all, but argues that this is not an excuse for Mickey having eaten it all when there was not a lot left.
Prove that Minnie is right by writing down a line of a truth table which shows that
$$( c \Rightarrow \sim e ) \Leftrightarrow ( \sim c \Rightarrow e )$$
is false.
(You may produce the whole table if you wish, but you need to indicate a specific line of the table.)
Show that the following combinatorial circuit is modelling an implication statement. Say what that statement is, and prove that the circuit and the statement are equivalent.
\includegraphics[max width=\textwidth, alt={}, center]{c3a528e4-b5fe-4bff-a77e-e3199bb225a1-2_188_533_1272_767}
Express the following combinatorial circuit as an equivalent implication statement.
\includegraphics[max width=\textwidth, alt={}, center]{c3a528e4-b5fe-4bff-a77e-e3199bb225a1-2_314_835_1599_616}
Explain why \(( \sim \mathrm { p } \wedge \sim \mathrm { q } ) \Rightarrow \mathrm { r }\), together with \(\sim \mathrm { r }\) and \(\sim \mathrm { q }\), give p .