***** Extra exercises *****

Notation: We use ~ for negation, & for conjunction, and | for disjunction


*** Validity and satisfiability ***

Determine whether each of the following are valid, contingent or unsatisfiable. No explanation necessary.

(a) ((p & ~q) => (r | q)) <=> (p & ~q & ~r)

(b) ~(a | b & ~c) | (a | b & ~c)

(c) (q => p) & (~s => ~q) & ~(p => s) & q

(d) (rich=>happy) & rich & unhappy

(e) (careful=>happy) & (happy=>excited) & (excited=>~careful)


*** Translation ***

Translate the following English sentences into sentences of propositional logic.  Use only the logical constants we provide. For instance, the sentence, "It is either raining or snowing" with the given logical constants:
raining = "It is raining"
snowing = "It is snowing"
should be answered by:
raining | snowing

(a) "To become my girlfriend, you must be smart, pretty, and nice"

gf = "become my girlfriend"
s = "you are smart"
p = "you are pretty"
n = "you are nice"

(b) "Unless I go to Korea or Japan, I will not be able to attend a World Cup match"

k = "I go to Korea"
j = "I go to Japan"
w = "I will be able to attend a World Cup match"

(c) I'm willing to help whenever you have a problem, except when I'm sleeping or very busy.

willHelp = "I'm willing to help"
haveProblem = "you have a problem"
asleep = "I'm sleeping"
busy = "I'm very busy"

(d) If the course is a breadth course then it may be waived only if a similar or more advanced course has been taken except if it was in another institution.

br = "the course is a breadth course"
wave = "the course may be waived"
similar = "a similar course has been taken"
advanced = "a more advanced course has been taken"
another = "it (the alternative course) was in another institution"


*** Entailment ***

Let Gamma and Delta be sets of sentences in propositional logic, and let phi and psi be individual sentences in propositional logic. State whether each of the following statements is true or false. No explanation necessary. On Notation: UNION is set union, and INT is set intersection. "Gamma |= psi" means Gamma (semantically) entails psi, and "Gamma |# psi" means Gamma does not entail psi.

(a) If Gamma |= phi and Delta |= phi then Gamma INT Delta |= phi

(b) If Gamma |= phi and Delta |= phi then Gamma UNION Delta |= phi

(c) If Gamma |= phi and Delta |# phi then Gamma UNION Delta |= phi

(d) If Gamma |# psi then Gamma |= ~psi.

(e) If Gamma |= ~psi then Gamma |# psi.