5. Construct a truth table for the given statement. PV -q Complete the truth table. pv -q (1) (2) T. T. (3) (4) T. (5) (6) F (7) (8) F (1) O F OT (2) O OT (3) O F OT (4) O T O F (5) O F OT (6) O T O F (7) O T O F (8) O F OT

do not handwrite this.

21. Let the following predicates be given. The domain is all computer science classes. I(r) = "x is interesting." U(x) = "x is useful." H(x, y) = “x is harder than y." M(x, y) = “x has more students than y." (a) Write the following statements in predicate logic. i. All interesting CS classes are useful. ii. There are some useful CS classes that are not interesting. iii. Every interesting CS class has more students than any non- interesting CS class. (b) Write the following predicate logic statement in everyday English. Don't just give a word-for-word translation; your sentence should make sense. (3)[I(x) A (Vy)(H(x,y)→ M(y, x))] (c) Formally negate the statement from part (b). Simplify your negation so that no quantifier lies within the scope of a negation. State which derivation rules you are using. (d) Give a translation of your negated statement in everyday English.

determine the truth value of each of the statements below. Justify your responses. (a) (3m € N) (n = N) 4m + 7n = 17. (b) (Vm € N)¬(³n € N) m² — n² = 7. (c) (Vm € N) (Vn € N) ((m ≤n) V (n ≤ m)) ⇒ (m = n).