Given two statements: p : 'One is less than seven' and q : 'Two is less than eight'. Which of the following represents the biconditional statement (p ↔ q)?

Let P and Q be two propositions. Which of the following logical expressions are equivalent? (I) P ∨ ¬Q (II) ¬(¬P ∧ Q) (III) (P ∧ Q) ∨ (P ∧ ¬Q) ∨ (¬P ∧ ¬Q) (IV) (P ∧ Q) ∨ (P ∧ ¬Q) ∨ (¬P ∧ Q)

Let θ(x, y, z) be the statement (x + y = z), where x, y, z are real numbers. Consider two quantified statements: (i) ∀x ∀y &exists;z θ(x, y, z) (ii) &exists;z ∀x ∀y θ(x, y, z) Which of the following is correct?