Final answer:
In mathematics, (i) the cardinality of F₃ is 1 since it is a subset of the set P, (ii) a formula φ of maximum length can be created using symbols from the given set and having a height of 4, (iii) the length of a propositional formula with 3 occurrences of binary connectives and 2 occurrences of negation is 5, and (iv) the length of a propositional formula with n occurrences of binary connectives and m occurrences of negation is n + m.
Step-by-step explanation:
(i) The cardinality of a set is the number of elements in the set. In this case, P={r} has only one element, which is 'r'. Therefore, the cardinality of F₃ would also be 1, since F₃ is a subset of P and can have at most one element.
(ii) To find a formula φ of maximum length where each symbol belongs to the set {p,∧, (}, and the height is 4, we can start by using all three symbols in different combinations at each level of the formula. For example, φ = (p∧p)∧(p∧p). This formula has a height of 4 and uses symbols from the given set.
(iii) The length of a propositional formula is the number of symbols it contains. Given that there are 3 occurrences of binary connectives (such as ∧) and 2 occurrences of the symbol for negation (¬), the length of the formula would be 3 + 2 = 5.
(iv) The length of a propositional formula that has n occurrences of binary connectives and m occurrences of the negation symbol (¬) can be calculated by adding those occurrences together. So, the length of the formula would be n + m.