Ex$.1.[1$pt$]$ Construct an example of two clauses that can be resolved together in two different ways giving two different outcomes.

Ex$.2.$ From "Horses are animals," it follows that "The head of a horse is the head of an animal." Demonstrate that this inference is valid by carrying out the following steps:

a $[1$pt$]$ Translate the premise and the conclusion into the language of first$-$order logic. Use three predicates: $\backslash (\backslash$operatorname$\{$HeadO$\}$ f$($h$,$ x$)\backslash )($meaning $"$ h is the head of x $"),\backslash (\backslash$operatorname$\{$Horse$\}($x$)\backslash ),$ and $\backslash (\backslash$operatorname$\{$Animal$\}($x$)\backslash ).$

b $[1$pt$]$ Negate the conclusion, and convert the premise and the negated conclusion into conjunctive normal form.

c $[1$pt$]$ Use resolution to show that the conclusion follows from the premise.

Ex$.3[3$pts$]$ Given the following Markov Decision Process. Do $5$ steps of value iteration algorithm assuming $\backslash (\backslash$gamma$=0\mathrm{.}5\backslash )$