ranslate the following English sentences into propositional formulas. Remember to

define what your propositional variables mean!

For example, if the sentence was $$The subject passed test $1$ and test $2.,$ a good answer

might look like this:

P : The subject passed test $1.$

Q: The subject passed test $2.$

P $$ Q

Note that your propositional variables must be defined as complete propositions, so you

should not just write:

P : test $1$

Q: test $2$

P $$ Q

Sometimes $($like in this case$),$ it is necessary to slightly paraphrase so that your definitions

are complete sentences.

Feel free to use abbreviations $($e$.$g$.,$P : The subj. passed test $1.)$ or to omit unimpor$-$

tant words $($e$.$g$.,$Q: Subj. passed test $2.)$ if you$$re feeling lazy.

Words like $$not$,$no$,$ or $$doesn$$t$$ should not appear in the definitions of your propo$-$

sitional variables; you should capture the negation in your translation using $.$

$($a$)$ I had good intentions and the highest hopes.

$($b$)$ The card must either have the same color as the previous card or the same number

as the previous card.

$($c$)$ I will eat my tie if the Cubs win the World Series.

$($d$)$ It smelled funny, but he ate it anyway.

$($e$)$ All prizes will be awarded provided that enough eligible entries are received.

$($f$)$ Participants were not timed on this task; however, most finished in less than $8$

minutes.

$($g$)$ The image is recovered if and only if the receiver has an authenticated secret key.

$($h$)$ The printer can hold a black ink cartridge or a color cartridge, but not both.

$($i$)$ Changing batteries is a tedious task if the nodes are deployed in remote and$/$or

hostile environments.

Hint: You$$ll need three separate atomic propositions to fully translate this sentence.

$($j$)$ The people will give up their arms only when the tyrant resigns and we get our

money back.

Hint: You$$ll need three separate atomic propositions to fully translate this sentence.

$($k$)$ The existence of a Lyapunov function is a necessary and sufficient condition for the

stability of the system.

Hint: For this proposition $($and the following three propositions$),$ you$$ll need to do

some paraphrasing to ensure that your atomic propositions are complete sentences.

$($l$)$ Passing the Turing Test is a sufficient condition for a being$$s intelligence.

$($m$)$ The absence of antibodies in the subject$$s body implies that they are not susceptible

to infection.

$($n$)$ Disciplinary knowledge and organizational skills are necessary conditions for effec$-$

tive teaching. Hint: You should have three separate atomic propositions here.

$($o$)$ Bonus: Your meal comes with one of the following: rice, mashed potatoes, or fried

plantains.

Hint: This one is trickier than you might think. I recommend testing your answer

out by building a truth table.

$($p$)$ Bonus: Ready or not, here I come.

Use the following definitions for your atomic variables:

R: I am ready.

C: Here I come.

$($q$)$ Bonus: One false move and the whole thing explodes.

Use the following definitions for your atomic variables:

M : Someone makes a false move.

X: The whole thing explodes.