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6.E: Chapter Six (Exercises)

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    Exercise \(\PageIndex{1}\): Basic Symbolization

    Symbolize the following ordinary language sentences into propositional logic using the five basic logic symbols, sentence letters, and parentheses/brackets.

    A. I used to go to the gym every day, but now my kids keep me busy.

    B. I used to go to the gym every day, but now if my kids are awake, I’m usually with them.

    C. I am the manager of this team, and what I say goes.

    D. I’m not the manager of this team, but my ideas should be taken seriously.

    E. I’m not going to marry you if you won’t even clean up your apartment when I come over.

    F. We need to get a divorce, or at the very least something needs to change.

    G. If I’m going to stay with you, then you need to treat me with respect and honor my wishes.

    H. No people here are warm enough.

    I. Either you light a fire, or I will light one instead

    J. Either you light a fire, or if you don’t, I will have to do it.

    K. I hate being on a diet whenever, but luckily only whenever it’s a calorie-restriction diet. (think “if, but luckily only if”).

    Exercise \(\PageIndex{2}\): More Complex Symbolization

    Symbolize the following into propositional logic. Be careful with different forms of “not both”, “neither nor”, “not x and not y”, “either not x or not y”, as well as different conditional forms like “A if B”, “If A then B”, “A only if B”, “only if A, will b”, and so on.

    A. I’m either not going to go out with you tonight, or not going hiking with you tomorrow.

    B. I go to the gym every day, but if I become a parent, I won’t go to the gym everyday.

    C. If and only if you clean your room will you get your allowance and be allowed to go out with friends later.

    D. I want you to come out with us, but I don’t want you to both feel uncomfortable and make others uncomfortable.

    E. Peace will finally reign only if those powerful folks who would make soldiers into expendable pawns and those who would make enemies into monsters are stripped of their power.

    F. If a judge is to rule fairly on constitutional matters, then neither may they have conflicts of interest, nor may they allow their training in constitutional law to lapse.

    G. Neither will we buy a new house if there isn’t one available nor will we move to a new apartment if the rent is too high.

    H. Either he leaves, or I do unless he apologizes and offers financial restitution.

    Exercise \(\PageIndex{3}\): Framing Words

    Use what you’ve learned about the framing words we use in English to mark off logical structures like “either...or...”, “if...then...”, and “both...and...” to symbolize the following complex English sentences:

    A. If you don’t learn grammar early on in life, then it becomes harder to both recognize grammatical structures and either succeed academically or at least reason logically.

    B. Terrible is the day that we surrender if we haven’t both given the battle our all and been just in our dealings with our opponents.

    C. If and only if we are both fair in our negotiations and we neither give the appearance of disrespect nor betray the fact that we are in fact loyal to the crown, will we either be successful in our negotiations or at least not be punished for failing to negotiate successfully.

    D. Provided we will not win this trade war, we will lose if and only if we either fail to compromise and fail to achieve some semblance of a draw, or we both give up too much in our negotiations and walk away with less than we had at the beginning.

    E. Only if we are able to win over the citizens’ hearts and win over their minds, will our regime-change operations be successful and will our sacrifices not have been in vain.

    Exercise \(\PageIndex{4}\): Main Operators

    Identify the Main Operator of each propositional logic formula. Many of these are from CC-BY 4.0 Intl License.

    A. (P \(\wedge\) Q)

    B. ((P \(\wedge\) R) \(\leftrightarrow\) (P \(\vee\) (Q \(\wedge\) \(\neg\)S)))

    C. ((Q \(\rightarrow\) (R \(\rightarrow\) S)) \(\rightarrow\) ((P \(\rightarrow\) Q) \(\rightarrow\) (P \(\rightarrow\) R)))

    D. (((R \(\leftrightarrow\) S) \(\vee\) (P \(\wedge\) \(\neg\)Q)) \(\vee\) (\(\neg\)R \(\wedge\) S))

    E. (P \(\wedge\) (P \(\wedge\) (P \(\wedge\) (P \(\wedge\) P))))

    F. [\(\neg\)[D \(\leftrightarrow\) \(\neg\)(X\(\rightarrow\) (Z \(\bullet\) Q))] \(\vee\) P]

    G. \(\neg\)(P \(\wedge\) (P \(\wedge\) (P \(\wedge\) (P \(\wedge\) P))))

    H. (\(\neg\)((R \(\leftrightarrow\) S) \(\vee\) \(\neg\)(P \(\wedge\) \(\neg\)Q)) \(\vee\) \(\neg\)(\(\neg\)R \(\wedge\) S))

    I. (Q \(\rightarrow\) (((R \(\rightarrow\) S) \(\rightarrow\) (P \(\rightarrow\) Q)) \(\rightarrow\) (P \(\rightarrow\) R)))

    (Careful: parentheses are different from problem C)

    Exercise \(\PageIndex{5}\): Computing Truth Values

    Compute the truth values of the following formulas, where A-L are all true and M-Z are all false.

    A. (A \(\wedge\) Q)

    B. ((P \(\wedge\) B) \(\leftrightarrow\) (L \(\vee\) (Q \(\wedge\) \(\neg\)S)))

    C. ((Q \(\rightarrow\) (R \(\rightarrow\) F)) \(\rightarrow\) ((D \(\rightarrow\) Q) \(\rightarrow\) (C \(\rightarrow\) R)))

    D. (((R \(\leftrightarrow\) S) \(\vee\) (P \(\wedge\) \(\neg\)Q)) \(\vee\) (\(\neg\)R \(\wedge\) S))

    E. (P \(\wedge\) (P \(\wedge\) (P \(\wedge\) (P \(\wedge\) P))))

    F. (P \(\wedge\) (B \(\wedge\) (C \(\wedge\) (D \(\wedge\) P))))

    G. [\(\neg\)[D \(\leftrightarrow\) \(\neg\)(X\(\rightarrow\) (Z \(\bullet\) Q))] \(\vee\) P]

    H. [\(\neg\)[D \(\leftrightarrow\) \(\neg\)(X\(\rightarrow\) (Z \(\bullet\) Q))] \(\vee\) K]

    I. \(\neg\)(P \(\wedge\) (P \(\wedge\) (P \(\wedge\) (P \(\wedge\) P))))

    J. (P \(\wedge\) \(\neg\)(B \(\wedge\) (\(\neg\)C \(\wedge\) \(\neg\)(D \(\wedge\) P))))

    K. (\(\neg\)((R \(\leftrightarrow\) S) \(\vee\) \(\neg\)(P \(\wedge\) \(\neg\)Q)) \(\vee\) \(\neg\)(\(\neg\)R \(\wedge\) S))

    L. (\(\neg\)((D \(\leftrightarrow\) S) \(\vee\) \(\neg\)(E \(\wedge\) \(\neg\)Q)) \(\vee\) \(\neg\)(\(\neg\)B \(\wedge\) S))

    M. ((Q \(\rightarrow\) ((R \(\rightarrow\) S)) \(\rightarrow\) (P \(\rightarrow\) Q) \(\rightarrow\) (P \(\rightarrow\) R)))

    N. ((B \(\rightarrow\) (R \(\rightarrow\) E)) \(\rightarrow\) ((A \(\rightarrow\) Q) \(\rightarrow\) (P \(\rightarrow\) R)))

    Exercise \(\PageIndex{6}\): Truth Tables Classify

    Build a Truth Table to Classify the following single propositions:

    A. \(\neg\)(P \(\wedge\) (P \(\wedge\) (P \(\wedge\) (P \(\wedge\) P))))

    B. (((R \(\leftrightarrow\) S) \(\vee\) (R \(\wedge\) \(\neg\)S)) \(\vee\) (\(\neg\)R \(\wedge\) S))

    C. ((P \(\wedge\) R) \(\leftrightarrow\) (P \(\vee\) (R \(\wedge\) \(\neg\)P)))

    D. [\(\neg\)[D \(\leftrightarrow\) \(\neg\)(E\(\rightarrow\) (F \(\bullet\) D))] \(\vee\) E]

    E. (((P \(\wedge\) Q) \(\vee\) R) \(\leftrightarrow\) ((P \(\vee\) R) \(\wedge\) (Q \(\vee\) R)))

    Exercise \(\PageIndex{7}\): Truth Tables Compare

    Build a truth table to compare the following sets of propositions. Many of these are from CC-BY 4.0 Intl License.

    A. [~(B\(\wedge\)E)\(\rightarrow\)I] / [(B\(\wedge\)E)\(\rightarrow\)~I]

    B. [\(\neg\)[Z \(\leftrightarrow\) \(\neg\)(X\(\rightarrow\) (Z \(\bullet\) Q))] \(\vee\) Q] / Z / (X \(\bullet\) Q)

    C. (P \(\rightarrow\) Q) / (~P \(\vee\) Q) / (~Q \(\rightarrow\) ~P)

    D. ((P \(\wedge\) R) / (P \(\vee\) (R \(\wedge\) \(\neg\)P)))

    E. \(\neg\)[D \(\leftrightarrow\) \(\neg\)(E\(\rightarrow\) (E \(\bullet\) D))] / [\(\neg\)D \(\leftrightarrow\) (\(\neg\)E\(\rightarrow\) (E \(\bullet\) D))]

    F. (P \(\rightarrow\) Q) / (Q \(\leftrightarrow\) R) / (P \(\rightarrow\) R)

    Exercise \(\PageIndex{8}\): Truth Tables Validity

    Build a truth table to test each of the following inferences for validity. Many of these are from CC-BY 4.0 Intl License.

    A. (P \(\vee\) R) // (\(\neg\)P \(\rightarrow\) \(\neg\)\(\neg\)R)

    B. P // (P \(\wedge\) \(\neg\)(P \(\rightarrow\) P))

    C. (P \(\rightarrow\) Q) // (Q \(\rightarrow\) P)

    D. A / (A \(\rightarrow\) B) // (B \(\rightarrow\) A)

    E. D / (C \(\rightarrow\) (C \(\rightarrow\) (C\(\rightarrow\) D)))

    F. \(\neg\)P / (P \(\leftrightarrow\) R) // (R \(\leftrightarrow\) \(\neg\)(P \(\rightarrow\) P))

    G. ((P \(\wedge\) Q) \(\leftrightarrow\) (P \(\vee\) R)) // ((P \(\wedge\) Q) \(\vee\) \(\neg\)(P \(\vee\) R))

    This page titled 6.E: Chapter Six (Exercises) is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Andrew Lavin via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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