You want to prove a theorem: In any group of 30 people, there must be at least five people who were all born on the same day of the week. Rewrite it using quantifiers. To prove it by contradiction you need a negation of this statement. Write it first as a logic statement with quantifiers and then as an English sentence.
Show that ¬∀x(P(x) → Q(x)) and ∃x(P(x)∧¬Q(x)) are logically equivalent.