Answer:
See steps below
Step-by-step explanation:
a)
[tex](p\rightarrow q)\rightarrow r\Leftrightarrow \neg(\neg p\vee q)\vee r[/tex] equivalence of (r implies s) with (not r or s)
[tex] \neg(\neg p\vee q)\vee r\Leftrightarrow (\neg \neg p\wedge \neg q)\vee r[/tex] De Morgan's Law
[tex] (\neg \neg p\wedge \neg q)\vee r\Leftrightarrow (p\wedge \neg q)\vee r[/tex] Double negation
[tex](p\wedge \neg q)\vee r\Leftrightarrow (p\vee r)\wedge (\neg q\vee r)[/tex] Distributive Law
The last expression is in CNF.
b)
i)
Modus Ponens states the following,
If (p implies q) is true and p is true, then q is true.
By watching the truth table of implication
[tex]\left[\begin{array}{ccc}p&q&p\rightarrow q\\T&T&T\\T&F&F\\F&T&T\\F&F&T\end{array}\right][/tex]
We can notice that the only row that satisfies
(p implies q) is true and p is true
is the first row, so q must be true.
ii)
Modus Tollens states that if (p implies q) is true and (not q) is true, then (not p) is true.
By watching the following truth table
[tex]\left[\begin{array}{ccccc}p&q&\neg p&\neg q&p\rightarrow q\\T&T&F&F&T\\T&F&F&T&F\\F&T&T&F&T\\F&F&T&T&T\end{array}\right][/tex]
We can notice that the only row that satisfies (p implies q) is true and (not q) is true, is the fourth row, so (not p) must be true.
