29 Use rules of inference to show that if Vx(P(x) V Q(x)), Vx(¬Q(x) v S(x)), Vx(R(x) → ¬S(x)), and 3X¬P(x) are true, then 3X¬R(x) is true.

(discrete structure)

Let the universe of discourse be NBA players (active and retired), C(x) mean "x has won an NBA championship", "T(xy) means "x is the same height as or taller than y", and k = Kareem Abdul-Jabbar. Consider the following two statements: Væ[C(x) → -T(x, k)] -3z[T(x, k) A C(x)]. The second statement claims Kareem is the tallest player to win an NBA championship, but the first does not. These equivalent statements both say Kareem is the tallest person to win the NBA championship. These equivalent statements say Kareem is taller than anyone who won the NBA Championship, but they don't say anything about whether Kareem won. The first statement claims Kareem is the tallest to win the NBA championship, but the second does not.

|x² + y², x² + y² < 4, y 2 0 0, fxx(x,y) = otherwise