Question 1

(i) Consider the different types of number systems: Binary, Decimal, Octal and Hexadecimal. From these types, identify the one to which each of the following numbers belongs?
(a) Your registration number
(b) Your mobile phone number
(c) Your birthday in numbers in the format of ddmmyyyy (for an e.g. if you are born in January 02nd in 2000 then the corresponding number is 02012000)

(ii) By showing all steps clearly, convert each number in part (i) to all other number systems.

Question 2

(i) Consider a propositional language where
• A ="Angelo comes to the party",
• B ="Bruno comes to the party",
• C ="Carlo comes to the party",
• D ="David comes to the party".

Formalize the following sentences:

(a) "If David comes to the party then Bruno and Carlo come too"
(b) "Carlo comes to the party only if Angelo and Bruno do not come"
(c) "David comes to the party if and only if Carlo comes and Angelo doesn't come"
(d) "If David comes to the party, then, if Carlo doesn't come then Angelo comes"
(e) "Carlo comes to the party provided that David doesn't come, but, if David comes, then Bruno doesn't come"
(f) "A necessary condition for Angelo coming to the party, is that, if Bruno and Carlo aren't coming, David comes"

(ii) Use the truth tables method to determine whether the formula (p ∧ ~q) → (p ∧ q) is a logical equivalence of the formula ~p

(iii) Let Loves(x,y) mean "x loves y", Traveler(x) mean "x is a traveler", City(x) mean "x is a city", Lives(x,y) mean "x lives in y"

(a) Translate the following proposition into the most natural equivalent statement in English.

∃x∀y∀z(city(x) ∧ Traveler(y) ∧ Lives(z, x)) → (Loves(y, x) ∧∼ Loves(z, x))

(b) Translate the statement: "No traveler loves the city they live in." into predicate logic.

* Need Only ONly the answers of the question number 2